To solve for \(n\) in the equation \(\sqrt{4^n} = 1024\), follow these steps:
1. Isolate the exponential term: Start with the given equation.
[tex]\[
\sqrt{4^n} = 1024
\][/tex]
2. Square both sides: Eliminate the square root by squaring both sides of the equation.
[tex]\[
(\sqrt{4^n})^2 = 1024^2
\][/tex]
Simplifying this, we get:
[tex]\[
4^n = 1024^2
\][/tex]
3. Calculate the right-hand side: Calculate \(1024^2\).
[tex]\[
1024^2 = 1048576
\][/tex]
4. Equate the expressions: Now we have the equation:
[tex]\[
4^n = 1048576
\][/tex]
5. Expressing \(4\) as a power of \(2\): Note that \(4\) can be written as \(2^2\). Therefore:
[tex]\[
4^n = (2^2)^n = 2^{2n}
\][/tex]
6. Rewrite the equation: Substitute \(2^{2n}\) for \(4^n\), giving us:
[tex]\[
2^{2n} = 1048576
\][/tex]
7. Expressing \(1048576\) as a power of \(2\): Calculate the power of 2 that equals \(1048576\). We find that:
[tex]\[
1048576 = 2^{20}
\][/tex]
So, the equation becomes:
[tex]\[
2^{2n} = 2^{20}
\][/tex]
8. Equate the exponents: If the bases are equal, the exponents must also be equal. Therefore:
[tex]\[
2n = 20
\][/tex]
9. Solve for \(n\): Divide both sides of the equation by 2:
[tex]\[
n = \frac{20}{2} = 10
\][/tex]
Hence, the value of \(n\) is:
[tex]\[
n = 10
\][/tex]
