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Simplify this $\sqrt{9^{16 x^{2}}}$ $sqrt(9^(16x^2))$ ?

3 Answers

Sep 9, 2015

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

Explanation:

Since $b^{2 m} = {(b^{m})}^{2}$ $b^(2m) = (b^m)^2$

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

$XXX = 9^{8 x^{2}}$ $color(white)("XXX") = 9^(8x^2)$

$XXX = {(9^{8})}^{x^{2}}$ $color(white)("XXX") = (9^8)^(x^2)$

$XXX = 43, 046, 721^{x^{2}}$ $color(white)("XXX")=43,046,721^(x^2)$

Sep 9, 2015

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

Explanation:

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

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

Sep 9, 2015

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

Explanation:

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

$\sqrt{x} = x^{\frac{1}{2}}$ $color(blue)(sqrt(x) = x^(1/2))" "$ and ${(x^{a})}^{b} = x^{a \cdot b}$ $" "color(blue)((x^a)^b = x^(a * b))$

In this case, you would get

$\sqrt{9^{16 x^{2}}} = {[9^{16 x^{2}}]}^{\frac{1}{2}} = 9^{16 x^{2} \cdot \frac{1}{2}} = 9^{8 x^{2}}$ $sqrt(9^(16x^2)) = [9^(16x^2)]^(1/2) = 9^(16x^2 * 1/2) = 9^(8x^2)$

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

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

Another approach you can use is

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

Alternatively, you can also use

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

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