Answer :
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]
[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]
