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Section11.3Simplifying Square Root Expressions

Expressions containing square roots can frequently be simplified if we identify the largest perfect square that divides evenly into the radicand (the number or expression under the radical sign). In this context the phrase "perfect squares" refers to the squares of the positive integers, the first few of which are \(1\text{,}\) \(4\text{,}\) \(9\text{,}\) \(16\text{,}\) \(25\text{,}\) etc. If you don't know the perfect squares at least through \(121\text{,}\) the practice problems associated with this lesson will be easier if you write them out.

The process of simplifying square roots is reliant upon the following rule.

\begin{equation*} \sqrt{a \cdot b}=\sqrt{a} \cdot \sqrt{b} \text{, } a \geq 0 \text{ and } b \geq 0 \end{equation*}

The process entails writing the original expression as the product of two square roots, one of which simplifies to an integer.

Consider \(\sqrt{28}\text{.}\) There are several factor pairs of \(28\text{,}\) but we are looking for a pair where one of the two numbers is a perfect square. The only pair that fits the bill is \(4\) and \(7\text{.}\) What follows are the steps entailed in simplifying the square root.

\begin{align*} \sqrt{28}\amp=\sqrt{\highlightr{4} \cdot \highlight{7}}\\ \amp=\highlightr{\sqrt{4}} \cdot \highlight{\sqrt{7}}\\ \amp=\highlightr{2}\highlight{\sqrt{7}} \end{align*}

Several more examples are shown below.

Example11.3.1

Simplify \(\sqrt{75}\text{.}\)

Solution
\begin{align*} \sqrt{75}\amp=\sqrt{\highlightr{25} \cdot \highlight{3}}\\ \amp=\highlightr{\sqrt{25}} \cdot \highlight{\sqrt{3}}\\ \amp=\highlightr{5}\highlight{\sqrt{3}} \end{align*}
Example11.3.2

Simplify \(\sqrt{40}\text{.}\)

Solution
\begin{align*} \sqrt{40}\amp=\sqrt{\highlightr{4} \cdot \highlight{10}}\\ \amp=\highlightr{\sqrt{4}} \cdot \highlight{\sqrt{10}}\\ \amp=\highlightr{2}\highlight{\sqrt{10}} \end{align*}
Example11.3.3

Simplify \(\sqrt{98}\text{.}\)

Solution
\begin{align*} \sqrt{98}\amp=\sqrt{\highlightr{49} \cdot \highlight{2}}\\ \amp=\highlightr{\sqrt{49}} \cdot \highlight{\sqrt{2}}\\ \amp=\highlightr{7}\highlight{\sqrt{2}} \end{align*}

For a square root expression to be fully simplified, we need to identify the greatest perfect square that divides evenly into the radicand. Consider, for example, \(\sqrt{48}\text{.}\) The work below is correct, but the square root is not fully simplified β€” can you see why?

\begin{align*} \sqrt{48}\amp=\sqrt{\highlightr{4} \cdot \highlight{12}}\\ \amp=\highlightr{\sqrt{4}} \cdot \highlight{\sqrt{12}}\\ \amp=\highlightr{2}\highlight{\sqrt{12}} \end{align*}

The issue is that \(12\) still has another perfect square factor of \(4\text{.}\) While we succeeded in finding a perfect square that evenly divides into \(48\text{,}\) we did not succeed in identifying the largest perfect square that evenly divides into \(48\text{.}\) The correct simplification follows.

\begin{align*} \sqrt{48}\amp=\sqrt{\highlightr{16} \cdot \highlight{3}}\\ \amp=\highlightr{\sqrt{16}} \cdot \highlight{\sqrt{3}}\\ \amp=\highlightr{4}\highlight{\sqrt{3}} \end{align*}

Sometimes the original expression contains factors other than the square root expression. We need to be careful to carry those factors along as we simplify the square root factor. Several examples follow.

Example11.3.4

Simplify \(8\sqrt{63}\)

Solution
\begin{align*} 8\sqrt{63}\amp=8 \cdot \sqrt{\highlightr{9} \cdot \highlight{7}}\\ \amp=8 \cdot \highlightr{\sqrt{9}} \cdot \highlight{\sqrt{7}}\\ \amp=8 \cdot \highlightr{3} \cdot \highlight{\sqrt{7}}\\ \amp=24\sqrt{7} \end{align*}
Example11.3.5

Simplify \(-\frac{2\sqrt{108}}{3}\)

Solution
\begin{align*} -\frac{2\sqrt{108}}{3}\amp=-\frac{2 \cdot \sqrt{\highlightr{36} \cdot \highlight{3}}}{3}\\ \amp=-\frac{2 \cdot \highlightr{\sqrt{36}} \cdot \highlight{\sqrt{3}}}{3}\\ \amp=-\frac{2 \cdot \highlightr{6} \cdot \highlight{\sqrt{3}}}{3}\\ \amp=-\frac{12 \cdot \sqrt{3}}{3}\\ \amp=-4\sqrt{3} \end{align*}
Example11.3.6

Simplify \(\frac{\sqrt{90}}{12}\)

Solution
\begin{align*} \frac{\sqrt{90}}{12}\amp=\frac{\sqrt{\highlightr{9} \cdot \highlight{10}}}{12}\\ \amp=\frac{\highlightr{\sqrt{9}} \cdot \highlight{\sqrt{10}}}{12}\\ \amp=\frac{\highlightr{3} \cdot \highlight{\sqrt{10}}}{12}\\ \amp=\frac{\sqrt{10}}{4} \end{align*}

Subsection11.3.1Exercises

Simplify each square root expression.

1

\(\sqrt{50}\)

Solution

\(\begin{aligned}[t] \sqrt{50}\amp=\sqrt{\highlightr{25} \cdot \highlight{2}}\\ \amp=\highlightr{\sqrt{25}} \cdot \highlight{\sqrt{2}}\\ \amp=\highlightr{5}\highlight{\sqrt{2}} \end{aligned}\)

2

\(\sqrt{20}\)

Solution

\(\begin{aligned}[t] \sqrt{20}\amp=\sqrt{\highlightr{4} \cdot \highlight{5}}\\ \amp=\highlightr{\sqrt{4}} \cdot \highlight{\sqrt{5}}\\ \amp=\highlightr{2}\highlight{\sqrt{5}} \end{aligned}\)

3

\(-\sqrt{27}\)

Solution

\(\begin{aligned}[t] -\sqrt{27}\amp=-\sqrt{\highlightr{9} \cdot \highlight{3}}\\ \amp=-\highlightr{\sqrt{9}} \cdot \highlight{\sqrt{3}}\\ \amp=-\highlightr{3}\highlight{\sqrt{3}} \end{aligned}\)

4

\(5\sqrt{72}\)

Solution

\(\begin{aligned}[t] 5\sqrt{72}\amp=5 \cdot \sqrt{\highlightr{36} \cdot \highlight{2}}\\ \amp=5 \cdot \highlightr{\sqrt{36}} \cdot \highlight{\sqrt{2}}\\ \amp=5 \cdot \highlightr{6} \cdot \highlight{\sqrt{2}}\\ \amp=30\sqrt{2} \end{aligned}\)

5

\(\frac{\sqrt{8}}{2}\)

Solution

\(\begin{aligned}[t] \frac{\sqrt{8}}{2}\amp=\frac{\sqrt{\highlightr{4} \cdot \highlight{2}}}{2}\\ \amp=\frac{\highlightr{\sqrt{4}} \cdot \highlight{\sqrt{2}}}{2}\\ \amp=\frac{\highlightr{2} \cdot \highlight{\sqrt{2}}}{2}\\ \amp=\sqrt{2} \end{aligned}\)

6

\(-9\sqrt{12}\)

Solution

\(\begin{aligned}[t] -9\sqrt{12}\amp=-9 \cdot \sqrt{\highlightr{4} \cdot \highlight{3}}\\ \amp=-9\highlightr{\sqrt{4}} \cdot \highlight{\sqrt{3}}\\ \amp=-9 \cdot \highlightr{2} \cdot \highlight{\sqrt{3}}\\ \amp=-18\sqrt{3} \end{aligned}\)