Question

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

Answer 1

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

Explanation:

Since `b^(2m) = (b^m)^2`

`sqrt(9^(16x^2)) = sqrt((9^(8x^2))^2)`

`color(white)("XXX") = 9^(8x^2)`

`color(white)("XXX") = (9^8)^(x^2)`

`color(white)("XXX")=43,046,721^(x^2)`

Answer 2

`3^(16x^2)` or `9^(8x^2)`

Explanation:

`sqrt(9^(16x^2)) = (9^(16x^2))^(1/2) = 9^((1/2)16x^2)`

`= (9^(1/2))^(16x^2) = 3^(16x^2)` OR `=9^((1/2*16)x^2) = 9^(8x^2)`

Answer 3

`3^(16x^2)`

Explanation:

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

`color(blue)(sqrt(x) = x^(1/2))" "` and `" "color(blue)((x^a)^b = x^(a * b))`

In this case, you would get

`sqrt(9^(16x^2)) = [9^(16x^2)]^(1/2) = 9^(16x^2 * 1/2) = 9^(8x^2)`

Since you know that `9 = 3^2`, you can rewrite this as

`9^(8x^2) = (3^2)^(8x^2) = 3^(16x^2)`

Another approach you can use is

`sqrt(9^(16x^2)) = sqrt((9^(8x^2))^2) = 9^(8x^2) = 3^(16x^2)`

Alternatively, you can also use

`sqrt(9^(16x^2)) = sqrt((9^(x^2))^16) = (9^(x^2))^8 = [(3^2)^(x^2)]^8 = 3^(16x^2)`