For \(p,q\neq 0\), what is the simplified form of \(\left(\frac{p^{-5}q^{4}}{p^{-1}q^{-2}}\right)^{-2}\)?
Answer and explanation
Correct answer: \(p^{8}q^{-12}\)
Using the quotient rule for exponents, \(\frac{p^{-5}q^4}{p^{-1}q^{-2}}=p^{-5-(-1)}q^{4-(-2)}=p^{-4}q^6\). Applying \((a^m)^n=a^{mn}\), we get \((p^{-4}q^6)^{-2}=p^8q^{-12}\), so option A is correct. Option B results from mishandling the signs while multiplying the exponents by the outer \(-2\). Exam tip: subtract exponents when dividing like bases, then multiply by the outside exponent.
Frequently asked questions
What is the correct answer to this question?
\(p^{8}q^{-12}\)
Why is this the correct answer?
Using the quotient rule for exponents, \(\frac{p^{-5}q^4}{p^{-1}q^{-2}}=p^{-5-(-1)}q^{4-(-2)}=p^{-4}q^6\). Applying \((a^m)^n=a^{mn}\), we get \((p^{-4}q^6)^{-2}=p^8q^{-12}\), so option A is correct. Option B results from mishandling the signs while multiplying the exponents by the outer \(-2\). Exam tip: subtract exponents when dividing like bases, then multiply by the outside exponent.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Operations on real numbers and the laws of exponents.
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