Mathematics » Introducing Decimals » Simplify and Use Square Roots

Simplifying Variable Expressions with Square Roots

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

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

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