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Simplify. Express your answer using positive exponents.\newlineq−3q−5⋅q−3\frac{q^{-3}}{q^{-5} \cdot q^{-3}}

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Full solution

Q. Simplify. Express your answer using positive exponents.\newlineq−3q−5⋅q−3\frac{q^{-3}}{q^{-5} \cdot q^{-3}}
  1. Identify operation for exponents: q−3/(q−5⋅q−3)q^{-3}/(q^{-5} \cdot q^{-3})\newlineIdentify the operation needed for the exponents when dividing and multiplying powers with the same base.
  2. Rewrite expression: q−3/(q−5⋅q−3)q^{-3}/(q^{-5} \cdot q^{-3})\newlineWhen dividing powers with the same base, we subtract the exponents in the numerator from the exponents in the denominator. When multiplying powers with the same base, we add the exponents.\newlineq−3/(q−5⋅q−3)q^{-3}/(q^{-5} \cdot q^{-3}) can be rewritten as q−3/q−5+−3q^{-3}/q^{-5 + -3}
  3. Perform addition in denominator: q−3/q(−5+−3)q^{-3}/q^{(-5 + -3)}\newlinePerform the addition in the denominator.\newlineq−3/q−8q^{-3}/q^{-8}
  4. Subtract exponents: q−3/q−8q^{-3}/q^{-8}\newlineNow subtract the exponent in the denominator from the exponent in the numerator.\newlineq−3−(−8)q^{-3} - (-8) which simplifies to q(−3+8)q^{(-3 + 8)}
  5. Perform addition: q(−3+8)q^{(-3 + 8)}\newlinePerform the addition.\newlineq5q^5

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