Simplifying Variable Expressions with Square Roots

Expressions with square root that we have looked at so far have not had any variables. What happens when we have to find a square root of a variable expression?

Consider \(\sqrt{9{x}^{2}},\) where \(x\ge 0.\) Can you think of an expression whose square is \(9{x}^{2}?\)

\(\begin{array}{ccc}\hfill {\left(?\right)}^{2}& =& 9{x}^{2}\hfill \\ \hfill {\left(3x\right)}^{2}& =& 9{x}^{2}\phantom{\rule{2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}\sqrt{9{x}^{2}}=3x\hfill \end{array}\)

When we use a variable in a square root expression, for our work, we will assume that the variable represents a non-negative number. In every example and exercise that follows, each variable in a square root expression is greater than or equal to zero.

Example

Simplify: \(\sqrt{{x}^{2}}.\)

Solution

Think about what we would have to square to get \({x}^{2}\). Algebraically, \({\left(?\right)}^{2}={x}^{2}\)

	\(\sqrt{{x}^{2}}\)
Since \({\left(x\right)}^{2}={x}^{2}\)	\(x\)

Example

Simplify: \(\sqrt{16{x}^{2}}.\)

Solution

	\(\sqrt{16{x}^{2}}\)
\(\text{Since}\phantom{\rule{0.2em}{0ex}}{\left(4x\right)}^{2}=16{x}^{2}\)	\(4x\)

Example

Simplify: \(-\sqrt{81{y}^{2}}.\)

Solution

	\(-\sqrt{81{y}^{2}}\)
\(\text{Since}\phantom{\rule{0.2em}{0ex}}{\left(9y\right)}^{2}=81{y}^{2}\)	\(-9y\)

Example

Simplify: \(\sqrt{36{x}^{2}{y}^{2}}.\)

Solution

	\(\sqrt{36{x}^{2}{y}^{2}}\)
\(\text{Since}\phantom{\rule{0.2em}{0ex}}{\left(6xy\right)}^{2}=36{x}^{2}{y}^{2}\)	\(6xy\)

This lesson is part of:

Introducing Decimals

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