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Answer :
To simplify the expression [tex]\(5^{-8} \times 5^4\)[/tex], we can use the property of exponents which states that [tex]\(a^m \times a^n = a^{m+n}\)[/tex]. This property allows us to simply add the exponents when multiplying the same base. Here’s how we can do it step by step:
1. Identify the Base and Exponents: In the expression [tex]\(5^{-8} \times 5^4\)[/tex], the base is 5, and the exponents are -8 and 4.
2. Apply the Exponent Rule: Add the exponents together:
[tex]\[
5^{-8 + 4} = 5^{-4}
\][/tex]
3. Simplify the Expression: Next, we interpret the result [tex]\(5^{-4}\)[/tex]. A negative exponent means that we take the reciprocal of the base raised to the positive exponent:
[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 Reciprocal: Thus, [tex]\(5^{-4}\)[/tex] simplifies to:
[tex]\[
\frac{1}{625}
\][/tex]
So, the simplified form of the expression [tex]\(5^{-8} \times 5^4\)[/tex] is [tex]\(\frac{1}{625}\)[/tex]. Therefore, the correct answer is:
A. [tex]\(\frac{1}{625}\)[/tex]
1. Identify the Base and Exponents: In the expression [tex]\(5^{-8} \times 5^4\)[/tex], the base is 5, and the exponents are -8 and 4.
2. Apply the Exponent Rule: Add the exponents together:
[tex]\[
5^{-8 + 4} = 5^{-4}
\][/tex]
3. Simplify the Expression: Next, we interpret the result [tex]\(5^{-4}\)[/tex]. A negative exponent means that we take the reciprocal of the base raised to the positive exponent:
[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 Reciprocal: Thus, [tex]\(5^{-4}\)[/tex] simplifies to:
[tex]\[
\frac{1}{625}
\][/tex]
So, the simplified form of the expression [tex]\(5^{-8} \times 5^4\)[/tex] is [tex]\(\frac{1}{625}\)[/tex]. Therefore, the correct answer is:
A. [tex]\(\frac{1}{625}\)[/tex]
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