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To simplify a square root expression that does not involve a fraction, we can use the following two rules:
Simplify each square root.
$\sqrt{{a}^{4}}.$ The exponent is even:
$\frac{4}{2}=2.$ The exponent on the square root is 2.
$\sqrt{{a}^{4}}={a}^{2}$
$\sqrt{{a}^{6}{b}^{10}}.$ Both exponents are even:
$\frac{6}{2}=3$ and
$\frac{10}{2}=5.$ The exponent on the square root of
${a}^{6}$ is 3. The exponent on the square root if
${b}^{10}$ is 5.
$\sqrt{{a}^{6}{b}^{10}}={a}^{3}{b}^{5}$
$\sqrt{{y}^{5}}.$ The exponent is odd:
${y}^{5}={y}^{4}y.$ Then
$\sqrt{{y}^{5}}=\sqrt{{y}^{4}y}=\sqrt{{y}^{4}}\sqrt{y}={y}^{2}\sqrt{y}$
$\begin{array}{ccccc}\sqrt{36{a}^{7}{b}^{11}{c}^{20}}& =& \sqrt{{6}^{2}{a}^{6}a{b}^{10}b{c}^{20}}& & {a}^{7}={a}^{6}a,\text{\hspace{0.17em}}{b}^{11}={b}^{10}b\\ & =& \sqrt{{6}^{2}{a}^{6}{b}^{10}{c}^{20}\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}ab}& & \begin{array}{l}\text{by\hspace{0.17em}the\hspace{0.17em}commutative\hspace{0.17em}property\hspace{0.17em}of\hspace{0.17em}multiplication.}\end{array}\\ & =& \sqrt{{6}^{2}{a}^{6}{b}^{10}{c}^{20}}\sqrt{ab}& & \begin{array}{l}\text{by\hspace{0.17em}the\hspace{0.17em}product\hspace{0.17em}property\hspace{0.17em}of\hspace{0.17em}square\hspace{0.17em}roots.}\end{array}\\ & =& 6{a}^{3}{b}^{5}{c}^{10}\sqrt{ab}& & \end{array}$
$\begin{array}{ccc}\sqrt{49{x}^{8}{y}^{3}{\left(a-1\right)}^{6}}& =& \sqrt{{7}^{2}{x}^{8}{y}^{2}y{\left(a-1\right)}^{6}}\\ & =& \sqrt{{7}^{2}{x}^{8}{y}^{2}{\left(a-1\right)}^{6}}\sqrt{y}\\ & =& 7{x}^{4}y{\left(a-1\right)}^{3}\sqrt{y}\end{array}$
$\sqrt{75}=\sqrt{25\text{\hspace{0.17em}}\xb73}=\sqrt{{5}^{2}\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}3}=\sqrt{{5}^{2}}\sqrt{3}=5\sqrt{3}$
Simplify each square root.
$\sqrt{81{a}^{12}{b}^{6}{c}^{38}}$
$9{a}^{6}{b}^{3}{c}^{19}$
$\sqrt{144{x}^{4}{y}^{80}{\left(b+5\right)}^{16}}$
$12{x}^{2}{y}^{40}{\left(b+5\right)}^{8}$
$\sqrt{27{a}^{3}{b}^{4}{c}^{5}{d}^{6}}$
$3a{b}^{2}{c}^{2}{d}^{3}\sqrt{3ac}$
$\sqrt{180{m}^{4}{n}^{15}{\left(a-12\right)}^{15}}$
$6{m}^{2}{n}^{7}{\left(a-12\right)}^{7}\sqrt{5n\left(a-12\right)}$
A square root expression is in simplified form if there are
The square root expressions $\sqrt{5a},\frac{4\sqrt{3xy}}{5},$ and $\frac{11{m}^{2}n\sqrt{a-4}}{2{x}^{2}}$ are in simplified form.
The square root expressions $\sqrt{\frac{3x}{8}},\sqrt{\frac{4{a}^{4}{b}^{3}}{5}},$ and $\frac{2y}{\sqrt{3x}}$ are not in simplified form.
Simplify each square root.
$$\sqrt{\frac{9}{25}}=\frac{\sqrt{9}}{\sqrt{25}}=\frac{3}{5}$$
$$\sqrt{\frac{3}{5}}=\frac{\sqrt{3}}{\sqrt{5}}=\frac{\sqrt{3}}{\sqrt{5}}\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\frac{\sqrt{5}}{\sqrt{5}}=\frac{\sqrt{15}}{5}$$
$$\sqrt{\frac{9}{8}}=\frac{\sqrt{9}}{\sqrt{8}}=\frac{\sqrt{9}}{\sqrt{8}}\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\frac{\sqrt{8}}{\sqrt{8}}=\frac{3\sqrt{8}}{8}=\frac{3\sqrt{4\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}2}}{8}=\frac{3\sqrt{4}\sqrt{2}}{8}=\frac{3\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}2\sqrt{2}}{8}=\frac{3\sqrt{2}}{4}$$
$$\sqrt{\frac{{k}^{2}}{{m}^{3}}}=\frac{\sqrt{{k}^{2}}}{\sqrt{{m}^{3}}}=\frac{k}{\sqrt{{m}^{3}}}=\frac{k}{\sqrt{{m}^{2}m}}=\frac{k}{\sqrt{{m}^{2}}\sqrt{m}}=\frac{k}{m\sqrt{m}}=\frac{k}{m\sqrt{m}}\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}\frac{\sqrt{m}}{\sqrt{m}}=\frac{k\sqrt{m}}{m\sqrt{m}\sqrt{m}}=\frac{k\sqrt{m}}{m\text{\hspace{0.17em}}\xb7\text{\hspace{0.17em}}m}=\frac{k\sqrt{m}}{{m}^{2}}$$
$$\begin{array}{ccc}\sqrt{{x}^{2}-8x+16}& =& \sqrt{{\left(x-4\right)}^{2}}\\ & =& x-4\end{array}$$
Simplify each square root.
$\sqrt{\frac{{y}^{4}}{{x}^{3}}}$
$\frac{{y}^{2}\sqrt{x}}{{x}^{2}}$
$\sqrt{\frac{32{a}^{5}}{{b}^{7}}}$
$\frac{4{a}^{2}\sqrt{2ab}}{{b}^{4}}$
For the following problems, simplify each of the radical expressions.
$\sqrt{20{a}^{2}}$
$\sqrt{27{y}^{6}}$
$\sqrt{{m}^{7}}$
$\sqrt{{y}^{17}}$
$\sqrt{49{x}^{13}}$
$\sqrt{64{a}^{7}{b}^{3}}$
$8\sqrt{9{a}^{4}{b}^{11}}$
$8\sqrt{25{y}^{3}}$
$\sqrt{32{m}^{8}}$
$\sqrt{12{y}^{13}}$
$\sqrt{48{p}^{11}{q}^{5}}$
$8\sqrt{108{x}^{21}{y}^{3}}$
$-6\sqrt{72{x}^{2}{y}^{4}{z}^{10}}$
$-\sqrt{{c}^{18}}$
$\sqrt{4{x}^{2}{y}^{2}{z}^{2}}$
$-\sqrt{16{x}^{4}{y}^{5}}$
$\sqrt{{m}^{6}{n}^{8}{p}^{12}{q}^{20}}$
${m}^{3}{n}^{4}{p}^{6}{q}^{10}$
$\sqrt{{r}^{2}}$
$\sqrt{\frac{1}{4}}$
$\sqrt{\frac{4}{25}}$
$\frac{5\sqrt{8}}{\sqrt{3}}$
$\sqrt{\frac{5}{6}}$
$\sqrt{\frac{3}{10}}$
$-\sqrt{\frac{2}{5}}$
$\sqrt{\frac{16{a}^{2}}{5}}$
$\sqrt{\frac{72{x}^{2}{y}^{3}}{5}}$
$\sqrt{\frac{5}{b}}$
$\sqrt{\frac{12}{{y}^{5}}}$
$\sqrt{\frac{49{x}^{2}{y}^{5}{z}^{9}}{25{a}^{3}{b}^{11}}}$
$\frac{7x{y}^{2}{z}^{4}\sqrt{abyz}}{5{a}^{2}{b}^{6}}$
$\sqrt{\frac{27{x}^{6}{y}^{15}}{{3}^{3}{x}^{3}{y}^{5}}}$
$\sqrt{{\left(a-7\right)}^{8}}$
$\sqrt{{\left(x+2\right)}^{2}{\left(x+1\right)}^{2}}$
$\sqrt{{\left(a-3\right)}^{4}{\left(a-1\right)}^{2}}$
${\left(a-3\right)}^{2}\left(a-1\right)$
$\sqrt{{\left(b+7\right)}^{8}{\left(b-7\right)}^{6}}$
$\sqrt{{b}^{2}+6b+9}$
$\sqrt{{\left({a}^{2}-2a+1\right)}^{4}}$
${\left(a-1\right)}^{4}$
$\sqrt{{\left({x}^{2}+2x+1\right)}^{12}}$
( [link] ) Solve the inequality $3\left(a+2\right)\le 2\left(3a+4\right)$
$a\ge -\frac{2}{3}$
(
[link] ) Graph the inequality
$6x\le 5\left(x+1\right)-6.$
( [link] ) Supply the missing words. When looking at a graph from left-to-right, lines with _______ slope rise, while lines with __________ slope fall.
positive; negative
( [link] ) Simplify the complex fraction $\frac{5+\frac{1}{x}}{5-\frac{1}{x}}.$
( [link] ) Simplify $\sqrt{121{x}^{4}{w}^{6}{z}^{8}}$ by removing the radical sign.
$11{x}^{2}{w}^{3}{z}^{4}$
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