1) $ \sqrt{(36)(9)} = $
2) $ \sqrt[3]{\frac{64}{216}} = $
3) $ \frac{\sqrt{126}}{\sqrt{14}} = $
4) $ \sqrt[3]{\sqrt{4096}} = $
5) $ \sqrt[3]{8x^{3}46z^{12}} = $
6) $ \frac{\sqrt[4]{-7^{4}}}{\sqrt{16x^{8}}} = $
7) $ \frac{1}{4}\sqrt{\frac{45x^{6}}{80w^{2}}} = $
8) $ \left(-\frac{27}{125}\right)^{\frac{1}{3}} = $
9) $ \left(\frac{168^{8}}{81x^{12}}\right)^{-\frac{1}{4}} = $
10) $ \left(\frac{a^{-5}}{\sqrt{32}}\right)^{-\frac{2}{5}} = $

Q: 1) $ \sqrt{(36)(9)} = $ 2) $ \sqrt[3]{\frac{64}{216}} = $ 3) $ \frac{\sqrt{126}}{\sqrt{14}} = $ 4) $ \sqrt[3]{\sqrt{4096}} = $ 5) $ \sqrt[3]{8x^{3}46z^{12}} = $ 6) $ \frac{\sqrt[4]{-7^{4}}}{\sqrt{16x^{8}}} = $ 7) $ \frac{1}{4}\sqrt{\frac{45x^{6}}{80w^{2}}} = $ 8) $ \left(-\frac{27}{125}\right)^{\frac{1}{3}} = $ 9) $ \left(\frac{168^{8}}{81x^{12}}\right)^{-\frac{1}{4}} = $ 10) $ \left(\frac{a^{-5}}{\sqrt{32}}\right)^{-\frac{2}{5}} = $

Let's solve these expressions step by step: 1) $\sqrt{(36)(9)}$ The product inside the square root is $36 \times 9 = 324$. $\sqrt{324} = 18$. 2) $\sqrt[3]{\frac{64}{216}}$ Simplify the fraction: $\frac{64}{216} = \frac{8}{27}$. $\sqrt[3]{\frac{8}{27}} = \frac{\sqrt[3]{8}}{\sqrt[3]{27}} = \frac{2}{3}$. 3) $\frac{\sqrt{126}}{\sqrt{14}}$ Simplify under the square roots: $\sqrt{\frac{126}{14}} = \sqrt{9} = 3$. 4) $\sqrt[3]{4096}$ $4096 = 16^3$. $\sqrt[3]{4096} = 16$. 5) $\sqrt[3]{8x^3y^6z^{12}}$ $\sqrt[3]{8} = 2$, $\sqrt[3]{x^3} = x$, $\sqrt[3]{y^6} = y^2$, $\sqrt[3]{z^{12}} = z^4$. Result: $2xy^2z^4$. 6) $\sqrt[4]{\frac{-7^4}{16x^8}}$ $\sqrt[4]{(-7)^4} = 7$ (since the fourth root negates the negative sign), $\sqrt[4]{16} = 2$, $\sqrt[4]{x^8} = x^2$. Result: $\frac{7}{2x^2}$. 7) $\sqrt[4]{\frac{45x^6}{80w^2}}$ Simplify inside: $\frac{45}{80} = \frac{9}{16}$. $\sqrt[4]{\frac{9}{16}} = \frac{3}{4}$, $\sqrt[4]{x^6} = x^{3/2}$, $\sqrt[4]{w^2} = w^{1/2}$. Result: $\frac{3x^{3/2}}{4w^{1/2}}$. 8) $\left(-\frac{27}{125}\right)^{1/3}$ $\left(-\frac{27}{125}\right)^{1/3} = \frac{-3}{5}$. 9) $\left(\frac{16^8}{81x^{12}}\right)^{-1/4}$ $\left(\frac{16^8}{81x^{12}}\right)^{-1/4} = \left(\frac{1}{16^8} \times \frac{81x^{12}}{1}\right)^{1/4}$ $=\left(\frac{81x^{12}}{16^8}\right)^{1/4}$. Simplify: $\frac{3x^3}{4^2} = \frac{3x^3}{16}$. 10) $\left(\frac{a^{-9}}{\sqrt{32}}\right)^{-2/5}$ $\sqrt{32} = 2\sqrt{16} = 4\sqrt{2}$. $\left(\frac{a^{-9}}{4\sqrt{2}}\right)^{-2/5} = \left(a^9 \times \frac{1}{4\sqrt{2}}\right)^{-2/5}$. Simplify further as needed based on context.

Question

Accepted Answer

Let's solve these expressions step by step:

1) $\sqrt{(36)(9)}$

The product inside the square root is $36 \times 9 = 324$.
   
   $\sqrt{324} = 18$.

2) $\sqrt[3]{\frac{64}{216}}$

Simplify the fraction: $\frac{64}{216} = \frac{8}{27}$.
   
   $\sqrt[3]{\frac{8}{27}} = \frac{\sqrt[3]{8}}{\sqrt[3]{27}} = \frac{2}{3}$.

3) $\frac{\sqrt{126}}{\sqrt{14}}$

Simplify under the square roots: $\sqrt{\frac{126}{14}} = \sqrt{9} = 3$.

4) $\sqrt[3]{4096}$

$4096 = 16^3$.
   
   $\sqrt[3]{4096} = 16$.

5) $\sqrt[3]{8x^3y^6z^{12}}$

$\sqrt[3]{8} = 2$, $\sqrt[3]{x^3} = x$, $\sqrt[3]{y^6} = y^2$, $\sqrt[3]{z^{12}} = z^4$.
   
   Result: $2xy^2z^4$.

6) $\sqrt[4]{\frac{-7^4}{16x^8}}$

$\sqrt[4]{(-7)^4} = 7$ (since the fourth root negates the negative sign), $\sqrt[4]{16} = 2$, $\sqrt[4]{x^8} = x^2$.
   
   Result: $\frac{7}{2x^2}$.

7) $\sqrt[4]{\frac{45x^6}{80w^2}}$

Simplify inside: $\frac{45}{80} = \frac{9}{16}$.
   
   $\sqrt[4]{\frac{9}{16}} = \frac{3}{4}$, $\sqrt[4]{x^6} = x^{3/2}$, $\sqrt[4]{w^2} = w^{1/2}$.
   
   Result: $\frac{3x^{3/2}}{4w^{1/2}}$.

8) $\left(-\frac{27}{125}\right)^{1/3}$

$\left(-\frac{27}{125}\right)^{1/3} = \frac{-3}{5}$.

9) $\left(\frac{16^8}{81x^{12}}\right)^{-1/4}$

$\left(\frac{16^8}{81x^{12}}\right)^{-1/4} = \left(\frac{1}{16^8} \times \frac{81x^{12}}{1}\right)^{1/4}$
   
   $=\left(\frac{81x^{12}}{16^8}\right)^{1/4}$.
   
   Simplify: $\frac{3x^3}{4^2} = \frac{3x^3}{16}$.

10) $\left(\frac{a^{-9}}{\sqrt{32}}\right)^{-2/5}$

$\sqrt{32} = 2\sqrt{16} = 4\sqrt{2}$.
   
   $\left(\frac{a^{-9}}{4\sqrt{2}}\right)^{-2/5} = \left(a^9 \times \frac{1}{4\sqrt{2}}\right)^{-2/5}$.
   
   Simplify further as needed based on context.

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