If \((x^{2}y^{-1})^{k}=x^{10}y^{-5}\), where \(x,y\neq 0\), what is the value of \(k\)?
Answer and explanation
Correct answer: 5
Using the exponent rule \((a^m)^n=a^{mn}\), we get \((x^{2}y^{-1})^k=x^{2k}y^{-k}\). Comparing the exponents of the same bases gives \(2k=10\) and \(-k=-5\), so both equations yield \(k=5\). Option 4 is incorrect because it would produce \(x^8y^{-4}\). Exam tip: multiply every exponent inside the bracket by the outside exponent before comparing like bases.
Frequently asked questions
What is the correct answer to this question?
5
Why is this the correct answer?
Using the exponent rule \((a^m)^n=a^{mn}\), we get \((x^{2}y^{-1})^k=x^{2k}y^{-k}\). Comparing the exponents of the same bases gives \(2k=10\) and \(-k=-5\), so both equations yield \(k=5\). Option 4 is incorrect because it would produce \(x^8y^{-4}\). Exam tip: multiply every exponent inside the bracket by the outside exponent before comparing like bases.
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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