Simplify Expressions with Roots and Rational Exponents
Learning Objectives
- Introduction to Roots
- Define and evaluate principal square roots
- Define and evaluate nth roots
- Estimate roots that are not perfect
- Radical Expressions and Rational Exponents
- Define and identify a radical expression
- Convert radicals to expressions with rational exponents
- Convert expressions with rational exponents to their radical equivalent
- Simplify Radical Expressions
- Simplify radical expressions using factoring
- Simplify radical expressions using rational exponents and the laws of exponents
- Define [latex]\sqrt{x^2}=|x|[/latex], and apply it when simplifying radical expressions
Square Roots
The symbol for the square root is called a radical symbol and looks like this: [latex]\sqrt{\,\,\,}[/latex]. The expression [latex] \sqrt{25}[/latex] is read “the square root of twenty-five” or “radical twenty-five.” The number that is written under the radical symbol is called the radicand. The following table shows different radicals and their equivalent written and simplified forms.Radical | Name | Simplified Form |
---|---|---|
[latex] \sqrt{36}[/latex] | “Square root of thirty-six” “Radical thirty-six” | [latex] \sqrt{36}=\sqrt{6\cdot 6}=6[/latex] |
[latex] \sqrt{100}[/latex] | “Square root of one hundred” “Radical one hundred” | [latex] \sqrt{100}=\sqrt{10\cdot 10}=10[/latex] |
[latex] \sqrt{225}[/latex] | “Square root of two hundred twenty-five” “Radical two hundred twenty-five” | [latex] \sqrt{225}=\sqrt{15\cdot 15}=15[/latex] |
[latex] \begin{array}{r}5\cdot 5=25\\-5\cdot -5=25\end{array}[/latex]
By definition, the square root symbol always means to find the positive root, called the principal root. So while [latex]5\cdot5[/latex] and [latex]−5\cdot−5[/latex] both equal 25, only 5 is the principal root. You should also know that zero is special because it has only one square root: itself (since [latex]0\cdot0=0[/latex]). In our first example we will show you how to use radical notation to evaluate principal square roots.Example
Find the principal root of each expression.- [latex]\sqrt{100}[/latex]
- [latex]\sqrt{16}[/latex]
- [latex]\sqrt{25+144}[/latex]
- [latex]\sqrt{49}-\sqrt{81}\\[/latex]
- [latex] -\sqrt{81}[/latex]
- [latex]\sqrt{-9}[/latex]
Answer:
- [latex]\sqrt{100}=10[/latex] because [latex]{10}^{2}=100[/latex]
- [latex]\sqrt{\sqrt{16}}=\sqrt{4}=2[/latex] because [latex]{4}^{2}=16[/latex] and [latex]{2}^{2}=4[/latex]
- Recall that square roots act as grouping symbols in the order of operations, so addition and subtraction must be performed first when they occur under a radical. [latex]\sqrt{25+144}=\sqrt{169}=13[/latex] because [latex]{13}^{2}=169[/latex]
- This problem is similar to the last one, but this time subtraction should occur after evaluating the root. Stop and think about why these two problems are different. [latex]\sqrt{49}-\sqrt{81}=7 - 9=-2[/latex] because [latex]{7}^{2}=49[/latex] and [latex]{9}^{2}=81[/latex]
-
The negative in front means to take the opposite of the value after you simplify the radical. [latex] -\sqrt{81}\\-\sqrt{9\cdot 9}[/latex]. The square root of 81 is 9. Then, take the opposite of 9. [latex]−(9)[/latex]
- [latex]\sqrt{-9}[/latex], we are looking for a number that when it is squared, returns [latex]-9[/latex]. We can try [latex](-3)^2[/latex], but that will give a positive result, and [latex]3^2[/latex] will also give a positive result. This leads to an important fact - you cannot find the square root of a negative number.
Domain of a Square Root [latex]\sqrt{-a}[/latex] is not defined for all real numbers, a. Therefore, [latex]\sqrt{a}[/latex] is defined for [latex]a\ge0[/latex]
Think About It
Does [latex]\sqrt{25}=\pm 5[/latex]? Write your ideas and a sentence to defend them in the box below before you look at the answer. [practice-area rows="1"][/practice-area]Answer: No. Although both [latex]{5}^{2}[/latex] and [latex]{\left(-5\right)}^{2}[/latex] are [latex]25[/latex], the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is [latex]\sqrt{25}=5[/latex].
Cube Roots
We know that [latex]5^2=25, \text{ and }\sqrt{25}=5[/latex] but what if we want to "undo" [latex]5^3=125, \text{ or }5^4=625[/latex]? We can use higher order roots to answer these questions. Suppose we know that [latex]{a}^{3}=8[/latex]. We want to find what number raised to the 3rd power is equal to 8. Since [latex]{2}^{3}=8[/latex], we say that 2 is the cube root of 8. In the next example we will evaluate the cube roots of some perfect cubes.Example
Evaluate the following:- [latex] \sqrt[3]{125}[/latex]
- [latex] \sqrt[3]{-8}[/latex]
- [latex] \sqrt[3]{27}[/latex]
Answer: 1. You can read this as “the third root of 125” or “the cube root of 125.” To evaluate this expression, look for a number that, when multiplied by itself two times (for a total of three identical factors), equals 125. [latex]\text{?}\cdot\text{?}\cdot\text{?}=125[/latex]. Since 125 ends in 5, 5 is a good candidate. [latex]5/cdot5/cdot5=125[/latex] 2. We want to find a number whose cube is 8. [latex]2\cdot2\cdot2=8[/latex] the cube root of 8 is 2. 3. We want to find a number whose cube is -8. We know 2 is the cube root of 8, so maybe we can try -2. [latex]-2\cdot{-2}\cdot{-2}=-8[/latex], so the cube root of -8 is -2. This is different from square roots because multiplying three negative numbers together results in a negative number.
Nth Roots
The cube root of a number is written with a small number 3, called the index, just outside and above the radical symbol. It looks like [latex] \sqrt[3]{{}}[/latex]. This little 3 distinguishes cube roots from square roots which are written without a small number outside and above the radical symbol. We can apply the same idea to any exponent and it's corresponding root. The nth root of [latex]a[/latex] is a number that, when raised to the nth power, gives [latex]a[/latex]. For example, [latex]3[/latex] is the 5th root of [latex]243[/latex] because [latex]{\left(3\right)}^{5}=243[/latex]. If [latex]a[/latex] is a real number with at least one nth root, then the principal nth root of [latex]a[/latex] is the number with the same sign as [latex]a[/latex] that, when raised to the nth power, equals [latex]a[/latex]. The principal nth root of [latex]a[/latex] is written as [latex]\sqrt[n]{a}[/latex], where [latex]n[/latex] is a positive integer greater than or equal to 2. In the radical expression, [latex]n[/latex] is called the index of the radical.Definition: Principal nth Root
If [latex]a[/latex] is a real number with at least one nth root, then the principal nth root of [latex]a[/latex], written as [latex]\sqrt[n]{a}[/latex], is the number with the same sign as [latex]a[/latex] that, when raised to the nth power, equals [latex]a[/latex]. The index of the radical is [latex]n[/latex].Example
Evaluate each of the following:- [latex]\sqrt[5]{-32}[/latex]
- [latex]\sqrt[4]{81}[/latex]
- [latex]\sqrt[8]{-1}[/latex]
Answer:
- [latex]\sqrt[5]{-32}[/latex] Factor 32, because [latex]{\left(-2\right)}^{5}=-32 \\ \text{ }[/latex]
- [latex]\sqrt[4]{81}[/latex]. Factoring can help, we know that [latex]9\cdot9=81[/latex] and we can further factor each 9: [latex]\sqrt[4]{81}=\sqrt[4]{3\cdot3\cdot3\cdot3}=\sqrt[4]{3^4}=3[/latex]
- [latex]\sqrt[8]{-1}[/latex], since we have an 8th root - which is even- with a negative number as the radicand, this root has no real number solutions. In other words, [latex]-1\cdot-1\cdot-1\cdot-1\cdot-1\cdot-1\cdot-1\cdot-1=+1[/latex]
Estimate Roots
An approach to handling roots that are not perfect (squares, cubes, etc.) is to approximate them by comparing the values to perfect squares, cubes, or nth roots. Suppose you wanted to know the square root of 17. Let’s look at how you might approximate it.Example
Estimate. [latex] \sqrt{17}[/latex]Answer: Think of two perfect squares that surround 17. 17 is in between the perfect squares 16 and 25. So, [latex] \sqrt{17}[/latex] must be in between [latex] \sqrt{16}[/latex] and [latex] \sqrt{25}[/latex]. Determine whether [latex] \sqrt{17}[/latex] is closer to 4 or to 5 and make another estimate.
[latex] \sqrt{16}=4[/latex] and [latex] \sqrt{25}=5[/latex]
Since 17 is closer to 16 than 25, [latex] \sqrt{17}[/latex] is probably about 4.1 or 4.2. Use trial and error to get a better estimate of [latex] \sqrt{17}[/latex]. Try squaring incrementally greater numbers, beginning with 4.1, to find a good approximation for [latex] \sqrt{17}[/latex].[latex]\left(4.1\right)^{2}[/latex]
[latex]\left(4.1\right)^{2}[/latex] gives a closer estimate than [latex](4.2)^{2}[/latex].[latex]4.1\cdot4.1=16.81\\4.2\cdot4.2=17.64[/latex]
Continue to use trial and error to get an even better estimate.[latex]4.12\cdot4.12=16.9744\\4.13\cdot4.13=17.0569[/latex]
Answer
[latex-display] \sqrt{17}\approx 4.12[/latex-display]Example
Approximate [latex] \sqrt[3]{30}[/latex] and also find its value using a calculator.Answer: Find the cubes that surround 30. 30 is inbetween the perfect cubes 27 and 81. [latex] \sqrt[3]{27}=3[/latex] and [latex] \sqrt[3]{81}=4[/latex], so [latex] \sqrt[3]{30}[/latex] is between 3 and 4. Use a calculator.
[latex]\sqrt[3]{30}\approx3.10723[/latex]
Answer
By approximation: [latex]3\ge\sqrt[3]{30}\le4[/latex] Using a calculator: [latex] \sqrt[3]{30}\approx3.10723[/latex]Radical Expressions and Rational Exponents
Square roots are most often written using a radical sign, like this, [latex] \sqrt{4}[/latex]. But there is another way to represent them. You can use rational exponents instead of a radical. A rational exponent is an exponent that is a fraction. For example, [latex] \sqrt{4}[/latex] can be written as [latex] {{4}^{\tfrac{1}{2}}}[/latex]. Can’t imagine raising a number to a rational exponent? They may be hard to get used to, but rational exponents can actually help simplify some problems. Writing radicals with rational exponents will come in handy when we discuss techniques for simplifying more complex radical expressions. Radical expressions are expressions that contain radicals. Radical expressions come in many forms, from simple and familiar, such as[latex] \sqrt{16}[/latex], to quite complicated, as in [latex] \sqrt[3]{250{{x}^{4}}y}[/latex]Write an expression with a rational exponent as a radical
Radicals and fractional exponents are alternate ways of expressing the same thing. In the table below we show equivalent ways to express radicals: with a root, with a rational exponent, and as a principal root.
Radical Form |
Exponent Form |
Principal Root |
---|---|---|
[latex] \sqrt{16}[/latex] | [latex] {{16}^{\tfrac{1}{2}}}[/latex] | 4 |
[latex] \sqrt{25}[/latex] | [latex] {{25}^{\tfrac{1}{2}}}[/latex] | 5 |
[latex] \sqrt{100}[/latex] | [latex] {{100}^{\tfrac{1}{2}}}[/latex] | 10 |
Radical Form |
Exponent Form |
Principal Root |
---|---|---|
[latex] \sqrt[3]{8}[/latex] | [latex] {{8}^{\tfrac{1}{3}}}[/latex] | 2 |
[latex] \sqrt[3]{8}[/latex] | [latex] {{125}^{\tfrac{1}{3}}}[/latex] | 5 |
[latex] \sqrt[3]{1000}[/latex] | [latex] {{1000}^{\tfrac{1}{3}}}[/latex] | 10 |
Radical Form |
Exponent Form |
---|---|
[latex] \sqrt{x}[/latex] | [latex] {{x}^{\tfrac{1}{2}}}[/latex] |
[latex] \sqrt[3]{x}[/latex] | [latex] {{x}^{\tfrac{1}{3}}}[/latex] |
[latex] \sqrt[4]{x}[/latex] | [latex] {{x}^{\tfrac{1}{4}}}[/latex] |
… | … |
[latex] \sqrt[n]{x}[/latex] | [latex] {{x}^{\tfrac{1}{n}}}[/latex] |
Example
Express [latex] {{(2x)}^{^{\frac{1}{3}}}}[/latex] in radical form.Answer: Rewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root.
[latex]\sqrt[3]{2x} [/latex]
The parentheses in [latex] {{\left( 2x \right)}^{\frac{1}{3}}}[/latex] indicate that the exponent refers to everything within the parentheses.Answer
[latex-display] {{(2x)}^{^{\frac{1}{3}}}}=\sqrt[3]{2x}[/latex-display]Example
Express [latex] 2{{x}^{^{\frac{1}{3}}}}[/latex] in radical form.Answer: Rewrite the expression with the fractional exponent as a radical. The denominator of the fraction determines the root, in this case the cube root.
[latex] 2\sqrt[3]{x}[/latex]
The exponent refers only to the part of the expression immediately to the left of the exponent, in this case x, but not the 2.Answer
[latex-display] 2{{x}^{^{\frac{1}{3}}}}=2\sqrt[3]{x}[/latex-display]Write a radical expression as an expression with a rational exponent
Example
Write [latex] \sqrt[4]{81}[/latex] as an expression with a rational exponent.Answer: The radical form [latex] \Large\sqrt[4]{{\,\,\,\,}}[/latex] can be rewritten as the exponent [latex] \frac{1}{4}[/latex]. Remove the radical and place the exponent next to the base. [latex-display] {{81}^{\frac{1}{4}}}[/latex-display]
Answer
[latex-display] \sqrt[4]{81}={{81}^{\frac{1}{4}}}[/latex-display]Example
Express [latex] 4\sqrt[3]{xy}[/latex] with rational exponents.Answer: Rewrite the radical using a rational exponent. The root determines the fraction. In this case, the index of the radical is 3, so the rational exponent will be [latex] \frac{1}{3}[/latex].
[latex] 4{{(xy)}^{\frac{1}{3}}}[/latex]
Since 4 is outside the radical, it is not included in the grouping symbol and the exponent does not refer to it.Answer
[latex-display] 4\sqrt[3]{xy}=4{{(xy)}^{\frac{1}{3}}}[/latex-display]Rational exponents whose numerator is not equal to one
All of the numerators for the fractional exponents in the examples above were 1. You can use fractional exponents that have numerators other than 1 to express roots, as shown below.
Radical |
Exponent |
---|---|
[latex] \sqrt{9}[/latex] | [latex]9^{\frac{1}{2}}[/latex] |
[latex] \sqrt[3]{{{9}^{2}}}[/latex] | [latex]9^{\frac{2}{3}}[/latex] |
[latex]\sqrt[4]{9^{3}}[/latex] | [latex]9^{\frac{3}{4}}[/latex] |
[latex]\sqrt[5]{9^{2}}[/latex] | [latex]9^{\frac{2}{5}}[/latex] |
… | … |
[latex]\sqrt[n]{9^{x}}[/latex] | [latex]9\frac{x}{n}[/latex] |
Writing Rational Exponents
Any radical in the form [latex]\sqrt[n]{a^{x}}[/latex] can be written using a fractional exponent in the form [latex]a^{\frac{x}{n}}[/latex].Example
Rewrite the radicals using a rational exponent, then simplify your result.- [latex]\sqrt[3]{{{a}^{6}}}[/latex]
- [latex]\sqrt[12]{16^3}[/latex]
Answer: 1.[latex]\sqrt[n]{a^{x}}[/latex] can be rewritten as [latex]a^{\frac{x}{n}}[/latex], so in this case [latex]n=3,\text{ and }x=6[/latex], therefore [latex-display]\sqrt[3]{{{a}^{6}}}={{a}^{\frac{6}{3}}}[/latex-display] Simplify the exponent. [latex-display]{{a}^{\frac{6}{3}}}={{a}^{2}}[/latex-display]
Answer
[latex-display] \sqrt[3]{{{a}^{6}}}={{a}^{2}}[/latex-display] 2. [latex]\sqrt[n]{a^{x}}[/latex] can be rewritten as [latex]a^{\frac{x}{n}}[/latex], so in this case [latex]n=12,\text{ and }x=3[/latex], therefore[latex]\sqrt[12]{16^3}={16}^{\frac{3}{12}}={16}^{\frac{1}{4}}[/latex]
Simplify the expression using rules for exponents.[latex]\begin{array}{ccc}16=2^4\\{16}^{\frac{1}{4}}={2^4}^{\frac{1}{4}}\\=2^{4\cdot\frac{1}{4}}\\=2^1=2\end{array}[/latex]
Answer
[latex-display]\sqrt[12]{16^3}=2[/latex-display]Example
Rewrite the expressions using a radical.- [latex]{x}^{\frac{2}{3}}[/latex]
- [latex]{5}^{\frac{4}{7}}[/latex]
Answer:
- [latex]{x}^{\frac{2}{3}}[/latex], the numerator is 2 and the denominator is 3, therefore we will have the third root of x squared, [latex]\sqrt[3]{x^2}[/latex]
- [latex]{5}^{\frac{4}{7}}[/latex], the numerator is 4 and the denominator is 7, so we will have the seventh root of 5 raised to the fourth power. [latex]\sqrt[7]{5^4}[/latex]
Simplify Radical Expressions
Radical expressions are expressions that contain radicals. Radical expressions come in many forms, from simple and familiar, such as[latex] \sqrt{16}[/latex], to quite complicated, as in [latex] \sqrt[3]{250{{x}^{4}}y}[/latex]. To simplify complicated radical expressions, we can use some definitions and rules from simplifying exponents. Recall the Product Raised to a Power Rule from when you studied exponents. This rule states that the product of two or more non-zero numbers raised to a power is equal to the product of each number raised to the same power. In math terms, it is written [latex]\left(ab\right)^{x}=a^{x}\cdot{b}^{x}[/latex]. So, for example, you can use the rule to rewrite [latex] {{\left( 3x \right)}^{2}}[/latex] as [latex] {{3}^{2}}\cdot {{x}^{2}}=9\cdot {{x}^{2}}=9{{x}^{2}}[/latex]. Now instead of using the exponent 2, let’s use the exponent [latex] \frac{1}{2}[/latex]. The exponent is distributed in the same way.[latex] {{\left( 3x \right)}^{\frac{1}{2}}}={{3}^{\frac{1}{2}}}\cdot {{x}^{\frac{1}{2}}}[/latex]
And since you know that raising a number to the [latex] \frac{1}{2}[/latex] power is the same as taking the square root of that number, you can also write it this way.[latex] \sqrt{3x}=\sqrt{3}\cdot \sqrt{x}[/latex]
Look at that—you can think of any number underneath a radical as the product of separate factors, each underneath its own radical.A Product Raised to a Power Rule or sometimes called The Square Root of a Product Rule
For any real numbers a and b, [latex] \sqrt{ab}=\sqrt{a}\cdot \sqrt{b}[/latex]. For example: [latex] \sqrt{100}=\sqrt{10}\cdot \sqrt{10}[/latex], and [latex] \sqrt{75}=\sqrt{25}\cdot \sqrt{3}[/latex]The square root of a product rule will help us simplify roots that aren't perfect, as is shown the following example.
Example
Simplify. [latex] \sqrt{63}[/latex]Answer: 63 is not a perfect square so we can use the square root of a product rule to simplify any factors that are perfect squares. Factor 63 into 7 and 9. [latex-display] \sqrt{7\cdot 9}[/latex-display] 9 is a perfect square, [latex]9=3^2[/latex], therefore we can rewrite the radicand. [latex-display] \sqrt{7\cdot {{3}^{2}}}[/latex-display] Using the Product Raised to a Power rule, separate the radical into the product of two factors, each under a radical. [latex-display] \sqrt{7}\cdot \sqrt{{{3}^{2}}}[/latex-display] Take the square root of [latex]3^{2}[/latex]. [latex-display] \sqrt{7}\cdot 3[/latex-display] Rearrange factors so the integer appears before the radical, and then multiply. (This is done so that it is clear that only the 7 is under the radical, not the 3.) [latex-display] 3\cdot \sqrt{7}[/latex-display] Answer [latex-display] \sqrt{63}=3\sqrt{7}[/latex-display]
[latex]x[/latex] | [latex]x^{2}[/latex] | [latex]\sqrt{x^{2}}[/latex] | [latex]\left|x\right|[/latex] |
---|---|---|---|
[latex]−5[/latex] | 25 | 5 | 5 |
[latex]−2[/latex] | 4 | 2 | 2 |
0 | 0 | 0 | 0 |
6 | 36 | 6 | 6 |
10 | 100 | 10 | 10 |
Taking the Square Root of a Radical Expression
When finding the square root of an expression that contains variables raised to a power, consider that [latex]\sqrt{x^{2}}=\left|x\right|[/latex]. Examples: [latex]\sqrt{9x^{2}}=3\left|x\right|[/latex], and [latex]\sqrt{16{{x}^{2}}{{y}^{2}}}=4\left|xy\right|[/latex]Example
Simplify. [latex] \sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}[/latex]Answer: Factor to find variables with even exponents. [latex-display] \sqrt{{{a}^{2}}\cdot a\cdot {{b}^{4}}\cdot b\cdot {{c}^{2}}}[/latex-display] Rewrite [latex]b^{4}[/latex] as [latex]\left(b^{2}\right)^{2}[/latex]. [latex-display] \sqrt{{{a}^{2}}\cdot a\cdot {{({{b}^{2}})}^{2}}\cdot b\cdot {{c}^{2}}}[/latex-display] Separate the squared factors into individual radicals. [latex-display] \sqrt{{{a}^{2}}}\cdot \sqrt{{{({{b}^{2}})}^{2}}}\cdot \sqrt{{{c}^{2}}}\cdot \sqrt{a\cdot b}[/latex-display] Take the square root of each radical. Remember that [latex] \sqrt{{{a}^{2}}}=\left| a \right|[/latex]. [latex-display] \left| a \right|\cdot {{b}^{2}}\cdot \left|{c}\right|\cdot \sqrt{a\cdot b}[/latex-display] Simplify and multiply. [latex-display] \left| ac \right|{{b}^{2}}\sqrt{ab}[/latex-display]
Answer
[latex-display] \sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}=\left| ac \right|{{b}^{2}}\sqrt{ab}[/latex-display]Analysis of the Solution
Why didn't we write [latex]b^2[/latex] as [latex]|b^2|[/latex]? Because when you square a number, you will always get a positive result, so the principal square root of [latex]\left(b^2\right)^2[/latex] will always be non-negative. One tip for knowing when to apply the absolute value after simplifying any even indexed root is to look at the final exponent on your variable terms. If the exponent is odd - including 1 - add an absolute value. This applies to simplifying any root with an even index, as we will see in later examples. In the following video you will see more examples of how to simplify radical expressions with variables. https://youtu.be/q7LqsKPoAKo We will show another example where the simplified expression contains variables with both odd and even powers.Example
Simplify. [latex] \sqrt{9{{x}^{6}}{{y}^{4}}}[/latex]Answer: Factor to find identical pairs.
[latex] \sqrt{3\cdot 3\cdot {{x}^{3}}\cdot {{x}^{3}}\cdot {{y}^{2}}\cdot {{y}^{2}}}[/latex]
Rewrite the pairs as perfect squares.[latex] \sqrt{{{3}^{2}}\cdot {{\left( {{x}^{3}} \right)}^{2}}\cdot {{\left( {{y}^{2}} \right)}^{2}}}[/latex]
Separate into individual radicals.[latex] \sqrt{{{3}^{2}}}\cdot \sqrt{{{\left( {{x}^{3}} \right)}^{2}}}\cdot \sqrt{{{\left( {{y}^{2}} \right)}^{2}}}[/latex]
Simplify.[latex] 3{{x}^{3}}{{y}^{2}}[/latex]
Because x has an odd power, we will add the absolute value for our final solution.
[latex] 3|{{x}^{3}}|{{y}^{2}}[/latex]
Answer
[latex-display] \sqrt{9{{x}^{6}}{{y}^{4}}}=3|{{x}^{3}}|{y}[/latex-display]Example
Simplify. [latex] {{(36{{x}^{4}})}^{\frac{1}{2}}}[/latex]Answer: Rewrite the expression with the fractional exponent as a radical.
[latex] \sqrt{36{{x}^{4}}}[/latex]
Find the square root of both the coefficient and the variable.[latex]\begin{array}{r} \sqrt{{{6}^{2}}\cdot {{x}^{4}}}\\\sqrt{{{6}^{2}}}\cdot \sqrt{{{x}^{4}}}\\\sqrt{{{6}^{2}}}\cdot \sqrt{{{({{x}^{2}})}^{2}}}\\6\cdot{x}^{2}\end{array}[/latex]
Answer
[latex-display] {{(36{{x}^{4}})}^{\frac{1}{2}}}=6{{x}^{2}}[/latex-display]Example
Simplify. [latex] \sqrt{49{{x}^{10}}{{y}^{8}}}[/latex]Answer: Look for squared numbers and variables. Factor 49 into [latex]7\cdot7[/latex], [latex]x^{10}[/latex] into [latex]x^{5}\cdot{x}^{5}[/latex], and [latex]y^{8}[/latex] into [latex]y^{4}\cdot{y}^{4}[/latex].
[latex] \sqrt{7\cdot 7\cdot {{x}^{5}}\cdot {{x}^{5}}\cdot {{y}^{4}}\cdot {{y}^{4}}}[/latex]
Rewrite the pairs as squares.[latex] \sqrt{{{7}^{2}}\cdot {{({{x}^{5}})}^{2}}\cdot {{({{y}^{4}})}^{2}}}[/latex]
Separate the squared factors into individual radicals.[latex] \sqrt{{{7}^{2}}}\cdot \sqrt{{{({{x}^{5}})}^{2}}}\cdot \sqrt{{{({{y}^{4}})}^{2}}}[/latex]
Take the square root of each radical using the rule that [latex] \sqrt{{{x}^{2}}}=x[/latex].[latex] 7\cdot {{x}^{5}}\cdot {{y}^{4}}[/latex]
Multiply.[latex] 7{{x}^{5}}{{y}^{4}}[/latex]
Answer
[latex-display] \sqrt{49{{x}^{10}}{{y}^{8}}}=7|{{x}^{5}}|{{y}^{4}}[/latex-display]Simplify cube roots
We can use the same techniques we have used for simplifying square roots to simplify higher order roots. For example to simplify a cube root, the goal is to find factors under the radical that are perfect cubes so that you can take their cube root. We no longer need to be concerned about whether we have identified the principal root since we are now finding cube roots. Focus on finding identical trios of factors as you simplify.Example
Simplify. [latex] \sqrt[3]{40{{m}^{5}}}[/latex]Answer: Factor 40 into prime factors. [latex-display] \sqrt[3]{5\cdot 2\cdot 2\cdot 2\cdot {{m}^{5}}}[/latex-display] Since you are looking for the cube root, you need to find factors that appear 3 times under the radical. Rewrite [latex] 2\cdot 2\cdot 2[/latex] as [latex] {{2}^{3}}[/latex]. [latex-display] \sqrt[3]{{{2}^{3}}\cdot 5\cdot {{m}^{5}}}[/latex-display] Rewrite [latex] {{m}^{5}}[/latex] as [latex] {{m}^{3}}\cdot {{m}^{2}}[/latex]. [latex-display] \sqrt[3]{{{2}^{3}}\cdot 5\cdot {{m}^{3}}\cdot {{m}^{2}}}[/latex-display] Rewrite the expression as a product of multiple radicals. [latex-display] \sqrt[3]{{{2}^{3}}}\cdot \sqrt[3]{5}\cdot \sqrt[3]{{{m}^{3}}}\cdot \sqrt[3]{{{m}^{2}}}[/latex-display] Simplify and multiply. [latex-display] 2\cdot \sqrt[3]{5}\cdot m\cdot \sqrt[3]{{{m}^{2}}}[/latex-display]
Answer
[latex-display] \sqrt[3]{40{{m}^{5}}}=2m\sqrt[3]{5{{m}^{2}}}[/latex-display]Example
Simplify. [latex] \sqrt[3]{-27{{x}^{4}}{{y}^{3}}}[/latex]Answer: Factor the expression into cubes. Separate the cubed factors into individual radicals. [latex-display]\begin{array}{r}\sqrt[3]{-1\cdot 27\cdot {{x}^{4}}\cdot {{y}^{3}}}\\\sqrt[3]{{{(-1)}^{3}}\cdot {{(3)}^{3}}\cdot {{x}^{3}}\cdot x\cdot {{y}^{3}}}\\\sqrt[3]{{{(-1)}^{3}}}\cdot \sqrt[3]{{{(3)}^{3}}}\cdot \sqrt[3]{{{x}^{3}}}\cdot \sqrt[3]{x}\cdot \sqrt[3]{{{y}^{3}}}\end{array}[/latex-display] Simplify the cube roots. [latex-display] -1\cdot 3\cdot x\cdot y\cdot \sqrt[3]{x}[/latex-display]
Answer
[latex-display] \sqrt[3]{-27{{x}^{4}}{{y}^{3}}}=-3xy\sqrt[3]{x}[/latex-display]Example
Simplify. [latex] \sqrt[3]{-24{{a}^{5}}}[/latex]Answer: Factor [latex]−24[/latex] to find perfect cubes. Here, [latex]−1[/latex] and 8 are the perfect cubes.
[latex] \sqrt[3]{-1\cdot 8\cdot 3\cdot {{a}^{5}}}[/latex]
Factor variables. You are looking for cube exponents, so you factor [latex]a^{5}[/latex] into [latex]a^{3}[/latex] and [latex]a^{2}[/latex].[latex] \sqrt[3]{{{(-1)}^{3}}\cdot {{2}^{3}}\cdot 3\cdot {{a}^{3}}\cdot {{a}^{2}}}[/latex]
Separate the factors into individual radicals.[latex] \sqrt[3]{{{(-1)}^{3}}}\cdot \sqrt[3]{{{2}^{3}}}\cdot \sqrt[3]{{{a}^{3}}}\cdot \sqrt[3]{3\cdot {{a}^{2}}}[/latex]
Simplify, using the property [latex] \sqrt[3]{{{x}^{3}}}=x[/latex].[latex] -1\cdot 2\cdot a\cdot \sqrt[3]{3\cdot {{a}^{2}}}[/latex]
This is the simplest form of this expression; all cubes have been pulled out of the radical expression.[latex] -2a\sqrt[3]{3{{a}^{2}}}[/latex]
Answer
[latex-display] \sqrt[3]{-24{{a}^{5}}}=-2a\sqrt[3]{3{{a}^{2}}}[/latex-display]Simplifying fourth roots
Now let's move to simplifying fourth degree roots. No matter what root you are simplifying, the same idea applies, find cubes for cube roots, powers of four for fourth roots, etc. Recall that when your simplified expression contains an even indexed radical and a variable factor with an odd exponent, you need to apply an absolute value.Example
Simplify. [latex] \sqrt[4]{81{{x}^{8}}{{y}^{3}}}[/latex]Answer: Rewrite the expression. [latex-display] \sqrt[4]{81}\cdot \sqrt[4]{{{x}^{8}}}\cdot \sqrt[4]{{{y}^{3}}}[/latex-display] Factor each radicand. [latex-display] \sqrt[4]{3\cdot 3\cdot 3\cdot 3}\cdot \sqrt[4]{{{x}^{2}}\cdot {{x}^{2}}\cdot {{x}^{2}}\cdot {{x}^{2}}}\cdot \sqrt[4]{{{y}^{3}}}[/latex-display] Simplify. [latex-display]\begin{array}{r}\sqrt[4]{{{3}^{4}}}\cdot \sqrt[4]{{{({{x}^{2}})}^{4}}}\cdot \sqrt[4]{{{y}^{3}}}\\3\cdot {{x}^{2}}\cdot \sqrt[4]{{{y}^{3}}}\end{array}[/latex-display]
Answer
[latex-display]\sqrt[4]{81x^{8}y^{3}}=3x^{2}\sqrt[4]{y^{3}} [/latex-display]Example
Simplify. [latex] \sqrt[4]{81{{x}^{8}}{{y}^{3}}}[/latex]Answer: Rewrite the radical using rational exponents. [latex-display] {{(81{{x}^{8}}{{y}^{3}})}^{\frac{1}{4}}}[/latex-display] Use the rules of exponents to simplify the expression. [latex-display] \begin{array}{r}{{81}^{\frac{1}{4}}}\cdot {{x}^{\frac{8}{4}}}\cdot {{y}^{\frac{3}{4}}}\\{{(3\cdot 3\cdot 3\cdot 3)}^{\frac{1}{4}}}{{x}^{2}}{{y}^{\frac{3}{4}}}\\{{({{3}^{4}})}^{\frac{1}{4}}}{{x}^{2}}{{y}^{\frac{3}{4}}}\\3{{x}^{2}}{{y}^{\frac{3}{4}}}\end{array}[/latex-display] Change the expression with the rational exponent back to radical form. [latex-display] 3{{x}^{2}}\sqrt[4]{{{y}^{3}}}[/latex-display]
Answer
[latex-display] \sqrt[4]{81{{x}^{8}}{{y}^{3}}}=3{{x}^{2}}\sqrt[4]{{{y}^{3}}}[/latex-display]Example
Simplify. [latex]\large\frac{10{{b}^{2}}{{c}^{2}}}{c\sqrt[3]{8{{b}^{4}}}}[/latex]Answer: Separate the factors in the denominator. [latex-display] \frac{10{{b}^{2}}{{c}^{2}}}{c\cdot \sqrt[3]{8}\cdot \sqrt[3]{{{b}^{4}}}}[/latex-display] Take the cube root of 8, which is 2. [latex-display] \frac{10{{b}^{2}}{{c}^{2}}}{c\cdot 2\cdot \sqrt[3]{{{b}^{4}}}}[/latex-display] Rewrite the radical using a fractional exponent. [latex-display] \frac{10{{b}^{2}}{{c}^{2}}}{c\cdot 2\cdot {{b}^{\frac{4}{3}}}}[/latex-display] Rewrite the fraction as a series of factors in order to cancel factors (see next step). [latex-display] \frac{10}{2}\cdot \frac{{{c}^{2}}}{c}\cdot \frac{{{b}^{2}}}{{{b}^{\frac{4}{3}}}}[/latex-display] Simplify the constant and c factors. [latex-display] 5\cdot c\cdot \frac{{{b}^{2}}}{{{b}^{\frac{4}{3}}}}[/latex-display] Use the rule of negative exponents, n-x=[latex] \frac{1}{{{n}^{x}}}[/latex], to rewrite [latex] \frac{1}{{{b}^{\tfrac{4}{3}}}}[/latex] as [latex] {{b}^{-\tfrac{4}{3}}}[/latex]. [latex-display] 5c{{b}^{2}}{{b}^{-\ \frac{4}{3}}}[/latex-display] Combine the b factors by adding the exponents. [latex-display] 5c{{b}^{\frac{2}{3}}}[/latex-display] Change the expression with the fractional exponent back to radical form. By convention, an expression is not usually considered simplified if it has a fractional exponent or a radical in the denominator. [latex-display] 5c\sqrt[3]{{{b}^{2}}}[/latex-display]
Answer
[latex-display] \frac{10{{b}^{2}}{{c}^{2}}}{c\sqrt[3]{8{{b}^{4}}}}=5c\sqrt[3]{{{b}^{2}}}[/latex-display]Summary
A radical expression is a mathematical way of representing the nth root of a number. Square roots and cube roots are the most common radicals, but a root can be any number. To simplify radical expressions, look for exponential factors within the radical, and then use the property [latex] \sqrt[n]{{{x}^{n}}}=x[/latex] if n is odd, and [latex] \sqrt[n]{{{x}^{n}}}=\left| x \right|[/latex] if n is even to pull out quantities. All rules of integer operations and exponents apply when simplifying radical expressions. The steps to consider when simplifying a radical are outlined below.Simplifying a radical
When working with exponents and radicals:- If n is odd, [latex] \sqrt[n]{{{x}^{n}}}=x[/latex].
- If n is even, [latex] \sqrt[n]{{{x}^{n}}}=\left| x \right|[/latex]. (The absolute value accounts for the fact that if x is negative and raised to an even power, that number will be positive, as will the nth principal root of that number.)
Summary
The square root of a number is the number which, when multiplied by itself, gives the original number. Principal square roots are always positive and the square root of 0 is 0. You can only take the square root of values that are greater than or equal to 0. The square root of a perfect square will be an integer. Other roots can be simplified by identifying factors that are perfect squares, cubes, etc. Nth roots can be approximated using trial and error or a calculator. Any radical in the form [latex]\sqrt[n]{a^{x}}[/latex] can be written using a fractional exponent in the form [latex]a^{\frac{x}{n}}[/latex]. Rewriting radicals using fractional exponents can be useful in simplifying some radical expressions. When working with fractional exponents, remember that fractional exponents are subject to all of the same rules as other exponents when they appear in algebraic expressions.Licenses & Attributions
CC licensed content, Original
- Simplify a Variety of Square Expressions (Simplify Perfectly). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Simplify Cube Roots (Perfect Cube Radicands). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Simplify Perfect Nth Roots. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Approximate a Square Root to Two Decimal Places Using Trial and Error. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
- Write Expressions Using Radicals and Rational Exponents. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Simplify Square Roots (Not Perfect Square Radicands). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Simplify Square Roots with Variables. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Simplify Cube Roots (Not Perfect Cube Radicands). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Simplify Nth Roots with Variables. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
- Simplify Radicals Using Rational Exponents. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. License: CC BY: Attribution.
CC licensed content, Shared previously
- Unit 16: Radical Expressions and Quadratic Equations, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology Located at: https://www.nroc.org/. License: CC BY: Attribution.
- Precalculus. Provided by: Open Stax Authored by: Abramson, Jay. Located at: https://cnx.org/contents/[email protected]:1/Preface. License: CC BY: Attribution. License terms: Download for free at : http://cnx.org/contents/[email protected]:1/Preface.