Certainly! Let's solve the given expression step-by-step.
Step 1: Understand the given expression
We start with the expression:
[tex]\[
\frac{7^2 \cdot 7^8}{7^4}
\][/tex]
Step 2: Simplify the numerator
To simplify the numerator, we use the property of exponents which states that when multiplying two powers with the same base, we add their exponents. So:
[tex]\[
7^2 \cdot 7^8 = 7^{2+8} = 7^{10}
\][/tex]
Step 3: Simplify the entire expression
Now, we have the simplified numerator as [tex]\(7^{10}\)[/tex]. The given expression becomes:
[tex]\[
\frac{7^{10}}{7^4}
\][/tex]
To simplify this, we use the property of exponents which states that when dividing two powers with the same base, we subtract the exponents. So:
[tex]\[
\frac{7^{10}}{7^4} = 7^{10-4} = 7^6
\][/tex]
Step 4: Identify the values of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]
According to the given format [tex]\(\frac{7^a}{7^4} = 7^b\)[/tex], we can compare:
- From the numerator, we see that:
[tex]\[
7^a = 7^{10} \Rightarrow a = 10
\][/tex]
- From the simplified exponent:
[tex]\[
7^6 \Rightarrow b = 6
\][/tex]
Therefore, the values are:
[tex]\[
a = 10, \quad b = 6
\][/tex]
Thus:
[tex]\[
\boxed{(a = 10, \quad b = 6)}
\][/tex]
