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Answer :
To simplify the expression [tex]\(5^{-8} \times 5^4\)[/tex], we use the laws of exponents. Here's a step-by-step explanation:
1. Understand the property being used: The property of exponents we use in this problem is [tex]\(a^m \times a^n = a^{m+n}\)[/tex]. This tells us that when we multiply like bases, we add the exponents.
2. Apply the property: In the expression [tex]\(5^{-8} \times 5^4\)[/tex], both terms have the same base, which is 5. So, we can add the exponents:
[tex]\[
5^{-8} \times 5^4 = 5^{-8 + 4} = 5^{-4}
\][/tex]
3. Simplify [tex]\(5^{-4}\)[/tex]: An exponent with a negative power, like [tex]\(5^{-4}\)[/tex], can be rewritten as the reciprocal of the base raised to the positive power. So,
[tex]\[
5^{-4} = \frac{1}{5^4}
\][/tex]
4. Calculate [tex]\(5^4\)[/tex]: Compute [tex]\(5^4\)[/tex]:
[tex]\[
5^4 = 5 \times 5 \times 5 \times 5 = 625
\][/tex]
5. Find the final answer: Substitute back into the expression for [tex]\(5^{-4}\)[/tex]:
[tex]\[
5^{-4} = \frac{1}{625}
\][/tex]
So, the simplified answer is [tex]\(\frac{1}{625}\)[/tex], which corresponds to option A.
1. Understand the property being used: The property of exponents we use in this problem is [tex]\(a^m \times a^n = a^{m+n}\)[/tex]. This tells us that when we multiply like bases, we add the exponents.
2. Apply the property: In the expression [tex]\(5^{-8} \times 5^4\)[/tex], both terms have the same base, which is 5. So, we can add the exponents:
[tex]\[
5^{-8} \times 5^4 = 5^{-8 + 4} = 5^{-4}
\][/tex]
3. Simplify [tex]\(5^{-4}\)[/tex]: An exponent with a negative power, like [tex]\(5^{-4}\)[/tex], can be rewritten as the reciprocal of the base raised to the positive power. So,
[tex]\[
5^{-4} = \frac{1}{5^4}
\][/tex]
4. Calculate [tex]\(5^4\)[/tex]: Compute [tex]\(5^4\)[/tex]:
[tex]\[
5^4 = 5 \times 5 \times 5 \times 5 = 625
\][/tex]
5. Find the final answer: Substitute back into the expression for [tex]\(5^{-4}\)[/tex]:
[tex]\[
5^{-4} = \frac{1}{625}
\][/tex]
So, the simplified answer is [tex]\(\frac{1}{625}\)[/tex], which corresponds to option A.
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