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In our work with simplifying square root expressions, we noted that
$$\sqrt{xy}=\sqrt{x}\sqrt{y}$$
Since this is an equation, we may write it as
$$\sqrt{x}\sqrt{y}=\sqrt{xy}$$
To multiply two square root expressions, we use the product property of square roots.
In practice, it is usually easier to simplify the square root expressions before actually performing the multiplication. To see this, consider the following product:
$$\sqrt{8}\sqrt{48}$$
We can multiply these square roots in
either of two ways:
Simplify then multiply.
$$\sqrt{4\xb72}\sqrt{16\xb73}=\left(2\sqrt{2}\right)\left(4\sqrt{3}\right)=2\xb74\sqrt{2\xb73}=8\sqrt{6}$$
Multiply then simplify.
$$\sqrt{8}\sqrt{48}=\sqrt{8\xb748}=\sqrt{384}=\sqrt{64\xb76}=8\sqrt{6}$$
Notice that in the second method, the expanded term (the third expression, $\sqrt{384}$ ) may be difficult to factor into a perfect square and some other number.
The preceding example suggests that the following rule for multiplying two square root expressions.
Find each of the following products.
$$\sqrt{3}\sqrt{6}=\sqrt{3\xb76}=\sqrt{18}=\sqrt{9\xb72}=3\sqrt{2}$$
$$\sqrt{8}\sqrt{2}=2\sqrt{2}\sqrt{2}=2\sqrt{2\xb72}=2\sqrt{4}=2\xb72=4$$
This product might be easier if we were to multiply first and then simplify.
$$\sqrt{8}\sqrt{2}=\sqrt{8\xb72}=\sqrt{16}=4$$
$$\sqrt{20}\sqrt{7}=\sqrt{4}\sqrt{5}\sqrt{7}=2\sqrt{5\xb77}=2\sqrt{35}$$
$$\begin{array}{lll}\sqrt{5{a}^{3}}\sqrt{27{a}^{5}}=(a\sqrt{5a})(3{a}^{2}\sqrt{3a})\hfill & =\hfill & 3{a}^{3}\sqrt{15{a}^{2}}\hfill \\ \hfill & =\hfill & 3{a}^{3}\xb7a\sqrt{15}\hfill \\ \hfill & =\hfill & 3{a}^{4}\sqrt{15}\hfill \end{array}$$
$$\begin{array}{lll}\sqrt{{(x+2)}^{7}}\sqrt{x-1}=\sqrt{{(x+2)}^{6}(x+2)}\sqrt{x-1}\hfill & =\hfill & {(x+2)}^{3}\sqrt{(x+2)}\sqrt{x-1}\hfill \\ \hfill & =\hfill & {(x+2)}^{3}\sqrt{(x+2)(x-1)}\hfill \\ \begin{array}{cccc}& & & \begin{array}{cccc}& & & \begin{array}{cccc}& & & \text{or}\end{array}\end{array}\end{array}\hfill & =\hfill & {(x+2)}^{3}\sqrt{{x}^{2}+x-2}\hfill \end{array}$$
Find each of the following products.
$\sqrt{x+4}\sqrt{x+3}$
$\sqrt{\left(x+4\right)\left(x+3\right)}$
$\sqrt{9{\left(k-6\right)}^{3}}\sqrt{{k}^{2}-12k+36}$
$3{\left(k-6\right)}^{2}\sqrt{k-6}$
$\sqrt{3}\left(\sqrt{2}+\sqrt{5}\right)$
$\sqrt{6}+\sqrt{15}$
$\sqrt{2a}\left(\sqrt{5a}-\sqrt{8{a}^{3}}\right)$
$a\sqrt{10}-4{a}^{2}$
$\sqrt{32{m}^{5}{n}^{8}}\left(\sqrt{2m{n}^{2}}-\sqrt{10{n}^{7}}\right)$
$8{m}^{3}{n}^{2}\sqrt{n}-8{m}^{2}{n}^{5}\sqrt{5m}$
$\sqrt{3}\sqrt{15}$
$\sqrt{20}\sqrt{3}$
$\sqrt{45}\sqrt{50}$
$\sqrt{7}\sqrt{7}$
$\sqrt{15}\sqrt{15}$
$\sqrt{80}\sqrt{20}$
$\sqrt{7}\sqrt{a}$
$\sqrt{10}\sqrt{h}$
$\sqrt{48}\sqrt{x}$
$\sqrt{200}\sqrt{m}$
$\sqrt{x}\sqrt{x}$
$\sqrt{h}\sqrt{h}$
$\sqrt{6}\sqrt{6}$
$\sqrt{m}\sqrt{m}$
$\sqrt{{a}^{2}}\sqrt{a}$
$\sqrt{{y}^{3}}\sqrt{y}$
$\sqrt{k}\sqrt{{k}^{6}}$
$\sqrt{{x}^{3}}\sqrt{{x}^{7}}$
$\sqrt{{y}^{7}}\sqrt{{y}^{9}}$
$\sqrt{{x}^{8}}\sqrt{{x}^{5}}$
$\sqrt{x+2}\sqrt{x-3}$
$\sqrt{\left(x+2\right)\left(x-3\right)}$
$\sqrt{a-6}\sqrt{a+1}$
$\sqrt{y+3}\sqrt{y-2}$
$\sqrt{\left(y+3\right)\left(y-2\right)}$
$\sqrt{h+1}\sqrt{h-1}$
$\sqrt{x+9}\sqrt{{\left(x+9\right)}^{2}}$
$\left(x+9\right)\sqrt{x+9}$
$\sqrt{y-3}\sqrt{{\left(y-3\right)}^{5}}$
$\sqrt{3{a}^{2}}\sqrt{15{a}^{3}}$
$3{a}^{2\text{\hspace{0.17em}}}\sqrt{5a}$
$\sqrt{2{m}^{4}{n}^{3}}\sqrt{14{m}^{5}n}$
$\sqrt{12{\left(p-q\right)}^{3}}\sqrt{3{\left(p-q\right)}^{5}}$
$6{\left(p-q\right)}^{4}$
$\sqrt{15{a}^{2}{\left(b+4\right)}^{4}}\sqrt{21{a}^{3}{\left(b+4\right)}^{5}}$
$\sqrt{125{m}^{5}{n}^{4}{r}^{8}}\sqrt{8{m}^{6}r}$
$10{m}^{5}{n}^{2}{r}^{4}\sqrt{10mr}$
$\sqrt{7{\left(2k-1\right)}^{11}{\left(k+1\right)}^{3}}\sqrt{14{\left(2k-1\right)}^{10}}$
$\sqrt{{x}^{6}}\sqrt{{x}^{2}}\sqrt{{x}^{9}}$
$\sqrt{2{a}^{4}}\sqrt{5{a}^{3}}\sqrt{2{a}^{7}}$
$2{a}^{7}\sqrt{5}$
$\sqrt{{x}^{n}}\sqrt{{x}^{n}}$
$\sqrt{{a}^{2n+5}}\sqrt{{a}^{3}}$
$\sqrt{2{m}^{3n+1}}\sqrt{10{m}^{n+3}}$
$2{m}^{2n+2}\sqrt{5}$
$\sqrt{75{\left(a-2\right)}^{7}}\sqrt{48a-96}$
$\sqrt{2}\left(\sqrt{8}+\sqrt{6}\right)$
$2\left(2+\sqrt{3}\right)$
$\sqrt{5}\left(\sqrt{3}+\sqrt{7}\right)$
$\sqrt{3}\left(\sqrt{x}+\sqrt{2}\right)$
$\sqrt{3x}+\sqrt{6}$
$\sqrt{11}\left(\sqrt{y}+\sqrt{3}\right)$
$\sqrt{8}\left(\sqrt{a}-\sqrt{3a}\right)$
$2\sqrt{2a}-2\sqrt{6a}$
$\sqrt{x}\left(\sqrt{{x}^{3}}-\sqrt{2{x}^{4}}\right)$
$\sqrt{y}\left(\sqrt{{y}^{5}}+\sqrt{3{y}^{3}}\right)$
${y}^{2}\left(y+\sqrt{3}\right)$
$\sqrt{8{a}^{5}}\left(\sqrt{2a}-\sqrt{6{a}^{11}}\right)$
$\sqrt{12{m}^{3}}\left(\sqrt{6{m}^{7}}-\sqrt{3m}\right)$
$6{m}^{2}\left({m}^{3}\sqrt{2}-1\right)$
$\sqrt{5{x}^{4}{y}^{3}}\left(\sqrt{8xy}-5\sqrt{7x}\right)$
( [link] ) Factor ${a}^{4}{y}^{4}-25{w}^{2}.$
$\left({a}^{2}{y}^{2}+5w\right)\left({a}^{2}{y}^{2}-5w\right)$
( [link] ) Find the slope of the line that passes through the points $\left(-5,4\right)$ and $\left(-3,4\right).$
(
[link] ) Perform the indicated operations:
$$\frac{15{x}^{2}-20x}{6{x}^{2}+x-12}\xb7\frac{8x+12}{{x}^{2}-2x-15}\xf7\frac{5{x}^{2}+15x}{{x}^{2}-25}$$
$\frac{4\left(x+5\right)}{{\left(x+3\right)}^{2}}$
( [link] ) Simplify $\sqrt{{x}^{4}{y}^{2}{z}^{6}}$ by removing the radical sign.
( [link] ) Simplify $\sqrt{12{x}^{3}{y}^{5}{z}^{8}}.$
$2x{y}^{2}{z}^{4}\sqrt{3xy}$
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