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Study Guides > ALGEBRA / TRIG I

Simplifying Square Roots with Variables

Learning Outcomes

  • Simplify square roots with variables
  • Recognize that by definition x2\sqrt{x^{2}} is always nonnegative
Radical expressions are expressions that contain radicals. Radical expressions come in many forms, from simple and familiar, such as16 \sqrt{16}, to quite complicated, as in 250x4y3 \sqrt[3]{250{{x}^{4}}y}. Using factoring, you can simplify these radical expressions, too.
Radical: of or going to the root or origin; fundamental: a radical difference Radical

Simplifying Square Roots

Radical expressions will sometimes include variables as well as numbers. Consider the expression 9x6 \sqrt{9{{x}^{6}}}. Simplifying a radical expression with variables is not as straightforward as the examples we have already shown with integers. Consider the expression x2 \sqrt{{{x}^{2}}}. This looks like it should be equal to x, right? Let’s test some values for x and see what happens. In the chart below, look along each row and determine whether the value of x is the same as the value of x2 \sqrt{{{x}^{2}}}. Where are they equal? Where are they not equal? After doing that for each row, look again and determine whether the value of x2 \sqrt{{{x}^{2}}} is the same as the value of x\left|x\right|.
xx x2x^{2} x2\sqrt{x^{2}} x\left|x\right|
5−5 2525 55 55
2−2 44 22 22
00 00 00 00
66 3636 66 66
1010 100100 1010 1010
Notice—in cases where x is a negative number, x2x\sqrt{x^{2}}\neq{x}! (This happens because the process of squaring the number loses the negative sign, since a negative times a negative is a positive.) However, in all cases x2=x\sqrt{x^{2}}=\left|x\right|. You need to consider this fact when simplifying radicals that contain variables, because by definition x2\sqrt{x^{2}} is always nonnegative.

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 x2=x\sqrt{x^{2}}=\left|x\right|. Examples: 9x2=3x\sqrt{9x^{2}}=3\left|x\right|, and 16x2y2=4xy\sqrt{16{{x}^{2}}{{y}^{2}}}=4\left|xy\right|
Let’s try it. The goal is to find factors under the radical that are perfect squares so that you can take their square root.

Example

Simplify. 9x6 \sqrt{9{{x}^{6}}}

Answer: Factor to find identical pairs. 33x3x3 \sqrt{3\cdot 3\cdot {{x}^{3}}\cdot {{x}^{3}}} Rewrite the pairs as perfect squares, note how we use the power rule for exponents to simplify x6x^6 into a square: x32{x^3}^2 32(x3)2 \sqrt{{{3}^{2}}\cdot {{\left( {{x}^{3}} \right)}^{2}}} Separate into individual radicals. 32(x3)2 \sqrt{{{3}^{2}}}\cdot \sqrt{{{\left( {{x}^{3}} \right)}^{2}}} Simplify, using the rule that x2=x \sqrt{{{x}^{2}}}=\left|x\right|. 3x3 3\left|{{x}^{3}}\right|

Answer

9x6=3x3 \sqrt{9{{x}^{6}}}=3\left|{{x}^{3}}\right|

Variable factors with even exponents can be written as squares. In the example above, x6=x3x3=x32 {{x}^{6}}={{x}^{3}}\cdot{{x}^{3}}={\left|x^3\right|}^{2} and y4=y2y2=(y2)2 {{y}^{4}}={{y}^{2}}\cdot{{y}^{2}}={\left(|y^2\right|)}^{2}.

Try It

[ohm_question]189413[/ohm_question]
Let’s try to simplify another radical expression.

Example

Simplify. 100x2y4 \sqrt{100{{x}^{2}}{{y}^{4}}}

Answer: Separate factors; look for squared numbers and variables. Factor 100 into 101010\cdot10.

1010x2y4 \sqrt{10\cdot 10\cdot {{x}^{2}}\cdot {{y}^{4}}}

Factor y4y^{4} into (y2)2\left(y^{2}\right)^{2}.

1010x2(y2)2 \sqrt{10\cdot 10\cdot {{x}^{2}}\cdot {{({{y}^{2}})}^{2}}}

Separate the squared factors into individual radicals.

102x2(y2)2 \sqrt{{{10}^{2}}}\cdot \sqrt{{{x}^{2}}}\cdot \sqrt{{{({{y}^{2}})}^{2}}}

Take the square root of each radical . Since you do not know whether x is positive or negative, use x\left|x\right| to account for both possibilities, thereby guaranteeing that your answer will be positive.

10xy210\cdot\left|x\right|\cdot{y}^{2}

Simplify and multiply.

10xy210\left|x\right|y^{2}

Answer

100x2y4=10xy2 \sqrt{100{{x}^{2}}{{y}^{4}}}=10\left| x \right|{{y}^{2}}

You can check your answer by squaring it to be sure it equals 100x2y4 100{{x}^{2}}{{y}^{4}}.

Example

Simplify. 49x10y8 \sqrt{49{{x}^{10}}{{y}^{8}}}

Answer: Look for squared numbers and variables. Factor 49 into 777\cdot7, x10x^{10} into x5x5x^{5}\cdot{x}^{5}, and y8y^{8} into y4y4y^{4}\cdot{y}^{4}. 77x5x5y4y4 \sqrt{7\cdot 7\cdot {{x}^{5}}\cdot{{x}^{5}}\cdot{{y}^{4}}\cdot{{y}^{4}}} Rewrite the pairs as squares. 72(x5)2(y4)2 \sqrt{{{7}^{2}}\cdot{{({{x}^{5}})}^{2}}\cdot{{({{y}^{4}})}^{2}}} Separate the squared factors into individual radicals. 72(x5)2(y4)2 \sqrt{7^2}\cdot\sqrt{({x^5})^2}\cdot\sqrt{({y^4})^2} Take the square root of each radical using the rule that x2=x \sqrt{{{x}^{2}}}=\left|x\right|. 7x5y4 7\cdot\left|{{x}^{5}}\right|\cdot{{y}^{4}} Multiply. 7x5y4 7\left|{{x}^{5}}\right|{{y}^{4}}

Answer

49x10y8=7x5y4 \sqrt{49{{x}^{10}}{{y}^{8}}}=7\left|{{x}^{5}}\right|{{y}^{4}}

You find that the square root of 49x10y8 49{{x}^{10}}{{y}^{8}} is 7x5y47\left|{{x}^{5}}\right|{{y}^{4}}. In order to check this calculation, you could square 7x5y47\left|{{x}^{5}}\right|{{y}^{4}}, hoping to arrive at 49x10y8 49{{x}^{10}}{{y}^{8}}. And, in fact, you would get this expression if you evaluated (7x5y4)2 {\left({7\left|{{x}^{5}}\right|{{y}^{4}}}\right)^{2}}. In the video that follows we show several examples of simplifying radicals with variables. https://youtu.be/q7LqsKPoAKo

Example

Simplify. a3b5c2 \sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}

Answer: Factor to find variables with even exponents. a2ab4bc2 \sqrt{{{a}^{2}}\cdot a\cdot {{b}^{4}}\cdot{b}\cdot{{c}^{2}}} Rewrite b4b^{4} as (b2)2\left(b^{2}\right)^{2}. a2a(b2)2bc2 \sqrt{{{a}^{2}}\cdot a\cdot {{({{b}^{2}})}^{2}}\cdot{ b}\cdot{{c}^{2}}} Separate the squared factors into individual radicals. a2(b2)2c2ab \sqrt{{{a}^{2}}}\cdot\sqrt{{{({{b}^{2}})}^{2}}}\cdot\sqrt{{{c}^{2}}}\cdot \sqrt{a\cdot b} Take the square root of each radical. Remember that a2=a \sqrt{{{a}^{2}}}=\left| a \right|. ab2cab \left| a \right|\cdot {{b}^{2}}\cdot\left|{c}\right|\cdot\sqrt{a\cdot b} Simplify and multiply. The entire quantity ab2c a{{b}^{2}}c can be enclosed in the absolute value sign because b2b^2 will be positive anyway. ab2cab \left| a{{b}^{2}}c \right|\sqrt{ab}

Answer

a3b5c2=ab2cab \sqrt{{{a}^{3}}{{b}^{5}}{{c}^{2}}}=\left| a{{b}^{2}}c\right|\sqrt{ab}

In the next section, we will explore cube roots, and use the methods we have shown here to simplify them. Cube roots are unique from square roots in that it is possible to have a negative number under the root, such as 1253\sqrt[3]{-125}.

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