To solve the equation [tex]\(2^{5x} \times 2^x = \sqrt[5]{32}\)[/tex], follow these steps:
1. Simplify the right side of the equation:
[tex]\[
\sqrt[5]{32}
\][/tex]
Recognize that 32 is a power of 2:
[tex]\[
32 = 2^5
\][/tex]
Then:
[tex]\[
\sqrt[5]{32} = \sqrt[5]{2^5}
\][/tex]
Using the property of exponents, [tex]\(\sqrt[n]{a^m} = a^{m/n}\)[/tex], this becomes:
[tex]\[
\sqrt[5]{2^5} = 2^{5/5} = 2^1 = 2
\][/tex]
2. Simplify the left side of the equation:
[tex]\[
2^{5x} \times 2^x
\][/tex]
Using the property of exponents where [tex]\(a^m \times a^n = a^{m+n}\)[/tex], combine the exponents:
[tex]\[
2^{5x} \times 2^x = 2^{5x + x} = 2^{6x}
\][/tex]
3. Set the simplified forms equal:
[tex]\[
2^{6x} = 2
\][/tex]
Recognize that [tex]\(2\)[/tex] can be written as [tex]\(2^1\)[/tex]:
[tex]\[
2^{6x} = 2^1
\][/tex]
4. Since the bases are the same, set the exponents equal to each other:
[tex]\[
6x = 1
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
[tex]\[
x = \frac{1}{6}
\][/tex]
Therefore, the value of [tex]\(x\)[/tex] is:
[tex]\[
x = \frac{1}{6}
\][/tex]
