To simplify the expression [tex]\( 8^{-7} \cdot 8^5 \)[/tex] and express it using only positive exponents, follow these steps:
1. Use the properties of exponents:
Recall that when multiplying exponential expressions with the same base, you can add the exponents. That is,
[tex]\[
a^m \times a^n = a^{m+n}
\][/tex]
Here, we have the base [tex]\(8\)[/tex] and the exponents [tex]\(-7\)[/tex] and [tex]\(5\)[/tex].
2. Add the exponents:
Add the exponents [tex]\(-7\)[/tex] and [tex]\(5\)[/tex]:
[tex]\[
-7 + 5 = -2
\][/tex]
3. Re-write the expression:
Combine the base [tex]\(8\)[/tex] with the new exponent:
[tex]\[
8^{-7} \cdot 8^5 = 8^{-2}
\][/tex]
4. Express using positive exponents:
To express [tex]\(8^{-2}\)[/tex] with a positive exponent, use the property of exponents that states [tex]\( a^{-n} = \frac{1}{a^n} \)[/tex]:
[tex]\[
8^{-2} = \frac{1}{8^2}
\][/tex]
Therefore, the simplified expression, using only positive exponents, is:
[tex]\[
\frac{1}{8^2}
\][/tex]
So, the final answer after simplifying [tex]\( 8^{-7} \cdot 8^5 \)[/tex] and expressing it with positive exponents is:
[tex]\[
\frac{1}{8^2}
\][/tex]
