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Apply the quotient rule for exponents

For nonzero bases, the quotient of powers with the same base is simplified by subtracting the exponent in the denominator from the exponent in the numerator: am/an=am−na^m/a^n=a^{m-n}. This reasoning connects division with inverse powers and supports simplifying algebraic expressions and scientific notation; the exponents are not divided, and more advanced extensions to arbitrary real or complex exponents are not included.

Detailed Explanation: Apply the quotient rule for exponents

When dividing powers with the same nonzero base, keep the base and subtract the denominator’s exponent from the numerator’s exponent:

aman=am−n\frac{a^m}{a^n}=a^{m-n}

The exponents are subtracted, not divided.

Example:

5753\frac{5^7}{5^3}

  1. The bases are the same: both are 55.

  2. Subtract the denominator’s exponent from the numerator’s exponent:

7−3=47-3=4

  1. Keep the base 55:

5753=57−3=54\frac{5^7}{5^3}=5^{7-3}=5^4

  1. If needed, evaluate:

54=6255^4=625

So,

5753=54=625\boxed{\frac{5^7}{5^3}=5^4=625}

Learn by doing: Apply the quotient rule for exponents

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    Exponents - Division - Positive by Positive to Positive

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