# Simplify this sqrt(9^(16x^2))  ?

Sep 9, 2015

$\sqrt{{9}^{16 {x}^{2}}} = {9}^{8 {x}^{2}} = 43 , 046 , {721}^{{x}^{2}}$
(assuming you only want the primary square root)

#### Explanation:

Since ${b}^{2 m} = {\left({b}^{m}\right)}^{2}$

$\sqrt{{9}^{16 {x}^{2}}} = \sqrt{{\left({9}^{8 {x}^{2}}\right)}^{2}}$

$\textcolor{w h i t e}{\text{XXX}} = {9}^{8 {x}^{2}}$

$\textcolor{w h i t e}{\text{XXX}} = {\left({9}^{8}\right)}^{{x}^{2}}$

$\textcolor{w h i t e}{\text{XXX}} = 43 , 046 , {721}^{{x}^{2}}$

Sep 9, 2015

${3}^{16 {x}^{2}}$ or ${9}^{8 {x}^{2}}$

#### Explanation:

$\sqrt{{9}^{16 {x}^{2}}} = {\left({9}^{16 {x}^{2}}\right)}^{\frac{1}{2}} = {9}^{\left(\frac{1}{2}\right) 16 {x}^{2}}$

$= {\left({9}^{\frac{1}{2}}\right)}^{16 {x}^{2}} = {3}^{16 {x}^{2}}$ OR $= {9}^{\left(\frac{1}{2} \cdot 16\right) {x}^{2}} = {9}^{8 {x}^{2}}$

Sep 9, 2015

${3}^{16 {x}^{2}}$

#### Explanation:

You can simplify this expression using various properties of radicals and exponents. For example, you know that

$\textcolor{b l u e}{\sqrt{x} = {x}^{\frac{1}{2}}} \text{ }$ and $\text{ } \textcolor{b l u e}{{\left({x}^{a}\right)}^{b} = {x}^{a \cdot b}}$

In this case, you would get

$\sqrt{{9}^{16 {x}^{2}}} = {\left[{9}^{16 {x}^{2}}\right]}^{\frac{1}{2}} = {9}^{16 {x}^{2} \cdot \frac{1}{2}} = {9}^{8 {x}^{2}}$

Since you know that $9 = {3}^{2}$, you can rewrite this as

${9}^{8 {x}^{2}} = {\left({3}^{2}\right)}^{8 {x}^{2}} = {3}^{16 {x}^{2}}$

Another approach you can use is

$\sqrt{{9}^{16 {x}^{2}}} = \sqrt{{\left({9}^{8 {x}^{2}}\right)}^{2}} = {9}^{8 {x}^{2}} = {3}^{16 {x}^{2}}$

Alternatively, you can also use

$\sqrt{{9}^{16 {x}^{2}}} = \sqrt{{\left({9}^{{x}^{2}}\right)}^{16}} = {\left({9}^{{x}^{2}}\right)}^{8} = {\left[{\left({3}^{2}\right)}^{{x}^{2}}\right]}^{8} = {3}^{16 {x}^{2}}$