To solve the equation [tex]\(\frac{5^6}{5^2} = a^b\)[/tex], let's follow these steps:
1. Simplify the left-hand side:
We start with the expression [tex]\(\frac{5^6}{5^2}\)[/tex]. Using the properties of exponents, specifically the quotient rule [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex], we can simplify:
[tex]\[
\frac{5^6}{5^2} = 5^{6-2} = 5^4
\][/tex]
2. Compare the simplified left-hand side to the right-hand side:
After simplifying, we have:
[tex]\[
5^4 = a^b
\][/tex]
In this equation, it's clear that [tex]\(a = 5\)[/tex] because the base on both sides must be the same.
3. Determine the value of [tex]\(b\)[/tex]:
Now we have:
[tex]\[
5^4 = 5^b
\][/tex]
Since the bases are the same, the exponents must be equal for the equation to hold true. Therefore:
[tex]\[
b = 4
\][/tex]
Hence, the value of [tex]\(b\)[/tex] is [tex]\(4\)[/tex].
