To find the value of [tex]\( x \)[/tex] that makes the expression [tex]\(\left(\sqrt[4]{7^5}\right)^x\)[/tex] equal to 7, we need to solve the equation:
[tex]\[
\left(\sqrt[4]{7^5}\right)^x = 7
\][/tex]
First, let's rewrite [tex]\(\sqrt[4]{7^5}\)[/tex] in a simpler form:
[tex]\[
\sqrt[4]{7^5} = (7^5)^{1/4}
\][/tex]
Using the property of exponents [tex]\((a^m)^n = a^{mn}\)[/tex], we have:
[tex]\[
(7^5)^{1/4} = 7^{5 \cdot \frac{1}{4}} = 7^{5/4}
\][/tex]
Thus, the equation becomes:
[tex]\[
(7^{5/4})^x = 7
\][/tex]
Again, using the property of exponents [tex]\((a^m)^n = a^{mn}\)[/tex], we can simplify:
[tex]\[
7^{(5/4)x} = 7^1
\][/tex]
Since the bases are the same, we can equate the exponents:
[tex]\[
\frac{5}{4}x = 1
\][/tex]
Now solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{4}{5}
\][/tex]
Therefore, the value of [tex]\( x \)[/tex] that makes the expression [tex]\(\left(\sqrt[4]{7^5}\right)^x\)[/tex] equal to 7 is:
[tex]\[
\boxed{\frac{4}{5}}
\][/tex]
