To simplify the expression \( 125^3 \div 75^3 \) using the Quotient Rule of Integer Exponents, we can first express the bases as powers of their prime factors.
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Factor \( 125 \) and \( 75 \):
- \( 125 = 5^3 \)
- \( 75 = 3 \times 5^2 \)
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Rewrite the expression: \[ 125^3 = (5^3)^3 = 5^{9} \] \[ 75^3 = (3 \times 5^2)^3 = 3^3 \times 5^{6} \]
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Now substituting this into the division: \[ 125^3 \div 75^3 = \frac{5^9}{3^3 \times 5^6} \]
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Applying the Quotient Rule of Integer Exponents (\( a^m \div a^n = a^{m-n} \)): \[ \frac{5^9}{5^6} = 5^{9-6} = 5^3 \]
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Thus, the expression becomes: \[ \frac{5^3}{3^3} = \frac{5^3}{27} \]
So, the simplified expression is \( \frac{5^3}{27} \).
Alternatively, expressing \( 5^3 \) in terms of a base raised to an exponent as requested, we can say:
- \( 5^3 = 125 \)
The answer choices seem to suggest different forms of numbers. If we solely focus on \( \frac{5^3}{27} \), this aligns with \( 5^{3} \) as the first element in the quotient.
If we have to select from those options, none of them directly represents \( \frac{5^3}{3^3} \).
However, if your specific choice is among the options mentioned:
- \( 5^{106} \) and \( 5^{53} \) do not apply, given our simplification shows \( 5^3 \).
If we were to provide a numeric answer without considering the fraction:
- \( 5^3 = 125 \)
In conclusion: Typically, the simplified form would be expressed as \( 5^3 \) or further reduced if chosen specifically. However, none from the options explicitly reveal \( \frac{5^3}{27} \). Selecting \( 5^{53} \) or \( 5^{106} \) wouldn’t apply as correct under direct evaluation.
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