Question
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Tutor-Verified Answer
Answer
\[
\frac{5^{6}}{5^{10}} = \frac{1}{5^{4}}
\]
Solution
1. Write the expression:
\[
\frac{5^{6}}{5^{10}}
\]
2. Apply the quotient rule for exponents, which states:
\[
\frac{a^{m}}{a^{n}} = a^{m-n}
\]
Thus,
\[
\frac{5^{6}}{5^{10}} = 5^{6-10}
\]
3. Compute the exponent:
\[
6-10 = -4
\]
Therefore,
\[
5^{6-10} = 5^{-4}
\]
4. Express the negative exponent as a reciprocal:
\[
5^{-4} = \frac{1}{5^{4}}
\]
So, the simplified result is
\[
\frac{5^{6}}{5^{10}} = \frac{1}{5^{4}}
\]
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Simplify this solution The Deep Dive
To simplify \( \frac{5^{6}}{5^{10}} \), you can use the property of exponents that states \( \frac{a^{m}}{a^{n}} = a^{m-n} \). So, in this case, subtract the exponent in the denominator from the exponent in the numerator: \( 6 - 10 = -4 \). This gives you \( 5^{-4} \), which is the final simplified form. Remember, \( 5^{-4} \) can also be expressed as \( \frac{1}{5^{4}} \) if you're looking for a positive exponent! Now, if you're ever puzzled about exponents, a common mistake is forgetting that a negative exponent indicates a reciprocal. Always take a moment to check whether you need to flip the fraction if you end up with a negative exponent!
