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Study Guides > Intermediate Algebra

Read: Define and Evaluate Roots

Learning Objectives

  • Define and evaluate principal square roots
  • Define and evaluate nth roots
  • Estimate roots that are not perfect
The most common root is the square root. First, we will define what square roots are,  and how you find the square root of a number. Then we will apply similar ideas to define and evaluate nth roots. Roots are the inverse of exponents, much like multiplication is the inverse of division. Recall how exponents are defined, and written; with an exponent, as words, and as repeated multiplication. Exponent: 32 {{3}^{2}}45 {{4}^{5}}x3 {{x}^{3}}xn {{x}^{\text{n}}} Name: “Three squared” or “Three to the second power”, “Four to the fifth power”, “x cubed”, “x to the nth power” Repeated Multiplication: 33 3\cdot 3,  44444 4\cdot 4\cdot 4\cdot 4\cdot 4,  xxx x\cdot x\cdot x,  xxx...xn times \underbrace{x\cdot x\cdot x...\cdot x}_{n\text{ times}}. Conversely,  when you are trying to find the square root of a number (say, 2525), you are trying to find a number that can be multiplied by itself to create that original number. In the case of 2525, you can find that 55=255\cdot5=25, so 55 must be the square root.

Square Roots

The symbol for the square root is called a radical symbol and looks like this:    \sqrt{\,\,\,}. The expression 25 \sqrt{25} 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 expression: square root of twenty-five is enclosed in a circle. The circle has an arrow pointing to it labeled: Radical expression. The square root symbol has an arrow pointing to it labeled: Radical. The number twenty-five has an arrow pointing to it labeled: Radicand. The following table shows different radicals and their equivalent written and simplified forms.
Radical Name Simplified Form
36 \sqrt{36} “Square root of thirty-six” “Radical thirty-six” 36=66=6 \sqrt{36}=\sqrt{6\cdot 6}=6
100 \sqrt{100} “Square root of one hundred” “Radical one hundred” 100=1010=10 \sqrt{100}=\sqrt{10\cdot 10}=10
225 \sqrt{225} “Square root of two hundred twenty-five” “Radical two hundred twenty-five” 225=1515=15 \sqrt{225}=\sqrt{15\cdot 15}=15
Consider 25 \sqrt{25} again. You may realize that there is another value that, when multiplied by itself, also results in 2525. That number is 5−5.

55=2555=25 \begin{array}{r}5\cdot 5=25\\-5\cdot -5=25\end{array}

By definition, the square root symbol always means to find the positive root, called the principal root. So while 555\cdot5 and 55−5\cdot−5 both equal 2525, only 55 is the principal root. You should also know that zero is special because it has only one square root: itself (since 00=00\cdot0=0). 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.
  1. 100\sqrt{100}
  2. 16\sqrt{16}
  3. 25+144\sqrt{25+144}
  4. 4981\sqrt{49}-\sqrt{81}
  5. 81 -\sqrt{81}
  6. 9\sqrt{-9}

Answer:

  1. 100=10\sqrt{100}=10 because 102=100{10}^{2}=100
  2. 16=4\sqrt{16}=4 because 42=16{4}^{2}=16
  3. 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. 25+144=169=13\sqrt{25+144}=\sqrt{169}=13 because 132=169{13}^{2}=169
  4. 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. 4981=79=2\sqrt{49}-\sqrt{81}=7 - 9=-2 because 72=49{7}^{2}=49 and 92=81{9}^{2}=81
  5. The negative in front means to take the opposite of the value after you simplify the radical. 8199 -\sqrt{81}\\-\sqrt{9\cdot 9}.  The square root of 8181 is 99. Then, take the opposite of 99. (9)−(9)

  6. 9\sqrt{-9}, we are looking for a number that when it is squared, returns 9-9. We can try (3)2(-3)^2, but that will give a positive result, and 323^2 will also give a positive result. This leads to an important fact -  you cannot find the square root of a negative number.

In the following video we present more examples of how to find a principle square root. https://youtu.be/2cWAkmJoaDQ The last example we showed leads to an important characteristic of square roots. You can only take the square root of values that are nonnegative.

Domain of a Square Root a\sqrt{-a} is not defined for all real numbers, a. Therefore, a\sqrt{a} is defined for a0a\ge0

Think About It

Does 25=±5\sqrt{25}=\pm 5? 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 52{5}^{2} and (5)2{\left(-5\right)}^{2} are 2525, the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is 25=5\sqrt{25}=5.

Cube Roots

We know that 52=25, and 25=55^2=25, \text{ and }\sqrt{25}=5 but what if we want to "undo" 53=125, or 54=6255^3=125, \text{ or }5^4=625? We can use higher order roots to answer these questions. Suppose we know that a3=8{a}^{3}=8. We want to find what number raised to the 33rd power is equal to 88. Since 23=8{2}^{3}=8, we say that 22 is the cube root of 88. In the next example we will evaluate the cube roots of some perfect cubes.

Example

Evaluate the following:
  1. 1253 \sqrt[3]{125}
  2. 83 \sqrt[3]{-8}
  3. 273 \sqrt[3]{27}

Answer: 1. You can read this as “the third root of 125125” or “the cube root of 125125.” To evaluate this expression, look for a number that, when multiplied by itself two times (for a total of three identical factors), equals 125125. ???=125\text{?}\cdot\text{?}\cdot\text{?}=125. Since 125125 ends in 5,55, 5 is a good candidate. 555=1255\cdot{5}\cdot{5}=125 2. We want to find a number whose cube is 8-8. We know 22 is the cube root of 88, so maybe we can try 2-2. 222=8-2\cdot{-2}\cdot{-2}=-8, so the cube root of 8-8 is 2-2. This is different from square roots because multiplying three negative numbers together results in a negative number 3. We want to find a number whose cube is 2727. 333=273\cdot{3}\cdot{3}=27 the cube root of 2727 is 33.

As we saw in the last example,there is one interesting fact about cube roots that is not true of square roots. Negative numbers can’t have real number square roots, but negative numbers can have real number cube roots! What is the cube root of 8−8? 83=2 \sqrt[3]{-8}=-2 because 222=8 -2\cdot -2\cdot -2=-8. Remember, when you are multiplying an odd number of negative numbers, the result is negative! Consider (1)33=1 \sqrt[3]{{{(-1)}^{3}}}=-1. In the following video we show more examples of finding a cube root. https://youtu.be/9Nh-Ggd2VJo

Nth Roots

The cube root of a number is written with a small number 33, called the index, just outside and above the radical symbol. It looks like 3 \sqrt[3]{{}}. This little 33 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 aa is a number that, when raised to the nth power, gives aa. For example, 33 is the 5th root of 243243 because (3)5=243{\left(3\right)}^{5}=243. If aa is a real number with at least one nth root, then the principal nth root of aa is the number with the same sign as aa that, when raised to the nth power, equals aa. The principal nth root of aa is written as an\sqrt[n]{a}, where nn is a positive integer greater than or equal to 22. In the radical expression, nn is called the index of the radical.

Definition: Principal nth Root

If aa is a real number with at least one nth root, then the principal nth root of aa, written as an\sqrt[n]{a}, is the number with the same sign as aa that, when raised to the nth power, equals aa. The index of the radical is nn.

Example

Evaluate each of the following:
  1. 325\sqrt[5]{-32}
  2. 814\sqrt[4]{81}
  3. 18\sqrt[8]{-1}

Answer:

  1. 325\sqrt[5]{-32} Factor 3232, because (2)5=32 {\left(-2\right)}^{5}=-32 \\ \text{ }
  2. 814\sqrt[4]{81}. Factoring can help, we know that 99=819\cdot9=81 and we can further factor each 99: 814=33334=344=3\sqrt[4]{81}=\sqrt[4]{3\cdot3\cdot3\cdot3}=\sqrt[4]{3^4}=3
  3. 18\sqrt[8]{-1}, since we have an 88th root - which is even- with a negative number as the radicand, this root has no real number solutions. In other words, 11111111=+1-1\cdot-1\cdot-1\cdot-1\cdot-1\cdot-1\cdot-1\cdot-1=+1

In the following video we show more examples of how to evaluate and nth root. https://youtu.be/vA2DkcUSRSk You can find the odd root of a negative number, but you cannot find the even root of a negative number. This means you can evaluate the radicals 813, 645 \sqrt[3]{-81},\ \sqrt[5]{-64}, and 21877 \sqrt[7]{-2187}, but you cannot evaluate the radicals 100, 164 \sqrt[{}]{-100},\ \sqrt[4]{-16}, or 2,5006 \sqrt[6]{-2,500}.

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 1717. Let’s look at how you might approximate it.

Example

Estimate. 17 \sqrt{17}

Answer: Think of two perfect squares that surround 17171717 is in between the perfect squares 1616 and 2525. So, 17 \sqrt{17} must be in between 16 \sqrt{16} and 25 \sqrt{25}. Determine whether 17 \sqrt{17} is closer to 44 or to 55 and make another estimate.

16=4 \sqrt{16}=4 and 25=5 \sqrt{25}=5

Since 1717 is closer to 1616 than 2525, 17 \sqrt{17} is probably about 4.14.1 or 4.24.2. Use trial and error to get a better estimate of 17 \sqrt{17}. Try squaring incrementally greater numbers, beginning with 4.14.1, to find a good approximation for 17 \sqrt{17}.

(4.1)2\left(4.1\right)^{2}

(4.1)2\left(4.1\right)^{2} gives a closer estimate than (4.2)2(4.2)^{2}.

4.14.1=16.814.24.2=17.644.1\cdot4.1=16.81\\4.2\cdot4.2=17.64

Continue to use trial and error to get an even better estimate.

4.124.12=16.97444.134.13=17.05694.12\cdot4.12=16.9744\\4.13\cdot4.13=17.0569

Answer

174.12 \sqrt{17}\approx 4.12

This approximation is pretty close. If you kept using this trial and error strategy you could continue to find the square root to the thousandths, ten-thousandths, and hundred-thousandths places, but eventually it would become too tedious to do by hand. For this reason, when you need to find a more precise approximation of a square root, you should use a calculator. Most calculators have a square root key () (\sqrt{{}}) that will give you the square root approximation quickly. On a simple 44-function calculator, you would likely key in the number that you want to take the square root of and then press the square root key. Try to find 17 \sqrt{17} using your calculator. Note that you will not be able to get an “exact” answer because 17 \sqrt{17} is an irrational number, a number that cannot be expressed as a fraction, and the decimal never terminates or repeats. To nine decimal positions, 17 \sqrt{17} is approximated as 4.1231056264.123105626. A calculator can save a lot of time and yield a more precise square root when you are dealing with numbers that aren’t perfect squares.

Example

Approximate 303 \sqrt[3]{30} and also find its value using a calculator.

Answer: Find the cubes that surround 3030. 3030 is in between the perfect cubes 2727 and 8181. 273=3 \sqrt[3]{27}=3 and 813=4 \sqrt[3]{81}=4, so 303 \sqrt[3]{30} is between 33 and 44. Use a calculator.

3033.10723\sqrt[3]{30}\approx3.10723

Answer

By approximation: 330343\ge\sqrt[3]{30}\le4 Using a calculator: 3033.10723 \sqrt[3]{30}\approx3.10723

The following video shows another example of how to estimate a square root. https://youtu.be/iNfalyW7olk

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 00 is 00. You can only take the square root of values that are greater than or equal to 00. 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.

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