Sure, let's solve the equation step by step.
Given equation:
[tex]\[
\sqrt{2^{x+5}} = 16
\][/tex]
1. Square both sides to eliminate the square root:
[tex]\[
(\sqrt{2^{x+5}})^2 = 16^2
\][/tex]
This simplifies to:
[tex]\[
2^{x+5} = 256
\][/tex]
2. Express 256 as a power of 2:
We know that [tex]\(256\)[/tex] is a power of 2. Specifically:
[tex]\[
256 = 2^8
\][/tex]
3. Set the exponents equal to each other: Since the bases are the same, the exponents must be equal. Therefore, we have:
[tex]\[
x + 5 = 8
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Subtract 5 from both sides:
[tex]\[
x + 5 - 5 = 8 - 5
\][/tex]
Which simplifies to:
[tex]\[
x = 3
\][/tex]
So the solution is:
[tex]\[
x = 3
\][/tex]
Additionally, from squaring both sides, we found that:
[tex]\[
2^{x+5} = 256
\][/tex] which confirms that:
[tex]\[
256
\][/tex] is the correct numerical value.
Thus, the given equation [tex]\( \sqrt{2^{x+5}} = 16 \)[/tex] is satisfied when [tex]\( x = 3 \)[/tex].
