To simplify the expression [tex]\(\frac{6^{x+2}-6^x}{7 \times 6^x}\)[/tex]:
1. First, recognize that [tex]\(6^{x+2}\)[/tex] can be rewritten using the properties of exponents. Specifically, we can write [tex]\(6^{x+2}\)[/tex] as [tex]\(6^x \times 6^2\)[/tex]. Therefore:
[tex]\[
6^{x+2} = 6^x \times 36
\][/tex]
2. Substitute this into the original expression:
[tex]\[
\frac{6^x \times 36 - 6^x}{7 \times 6^x}
\][/tex]
3. Factor out [tex]\(6^x\)[/tex] from the numerator:
[tex]\[
\frac{6^x (36 - 1)}{7 \times 6^x}
\][/tex]
4. Simplify the expression inside the parenthesis:
[tex]\[
36 - 1 = 35
\][/tex]
So the expression becomes:
[tex]\[
\frac{6^x \times 35}{7 \times 6^x}
\][/tex]
5. Since [tex]\(6^x\)[/tex] is in both the numerator and the denominator, it cancels out:
[tex]\[
\frac{35}{7}
\][/tex]
6. Finally, simplify the fraction:
[tex]\[
\frac{35}{7} = 5
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\boxed{5}
\][/tex]
