If a is a positive real number with a \(\neq 1\), and \(\frac{a^{2p+1}\cdot a^{p-3}}{a^{p+4}}=a^6\), what is the value of p?
Answer and explanation
Correct answer: 6
Using the laws of exponents, \(\frac{a^{2p+1}\cdot a^{p-3}}{a^{p+4}}=a^{(2p+1)+(p-3)-(p+4)}=a^{2p-6}\). Since \(a>0\) and \(a\neq1\), equal powers with the same base have equal exponents; hence \(2p-6=6\). Therefore, \(2p=12\) and \(p=6\), so option C is correct. Exam tip: add exponents when multiplying powers with the same base and subtract them when dividing.
Frequently asked questions
What is the correct answer to this question?
6
Why is this the correct answer?
Using the laws of exponents, \(\frac{a^{2p+1}\cdot a^{p-3}}{a^{p+4}}=a^{(2p+1)+(p-3)-(p+4)}=a^{2p-6}\). Since \(a>0\) and \(a\neq1\), equal powers with the same base have equal exponents; hence \(2p-6=6\). Therefore, \(2p=12\) and \(p=6\), so option C is correct. Exam tip: add exponents when multiplying powers with the same base and subtract them when dividing.
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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