If \(a>0\), \(a\ne1\), and \(\frac{a^{3p-2}\cdot a^{p+5}}{a^{2p-1}}=a^{10}\) holds for all such values of \(a\), what is the value of \(p\)?

3

By the laws of exponents, exponents are added when like bases are multiplied and subtracted when they are divided. Thus, the exponent on the left side is \((3p-2)+(p+5)-(2p-1)=2p+4\). Therefore, \(2p+4=10\), giving \(2p=6\) and hence \(p=3\). Option C can result from mishandling the subtraction of the denominator’s exponent. In an exam, remember to change the sign of the exponent in the denominator while simplifying.

This is a Class 10 Mathematics question. Chapter: Polynomials. Topic: Operations on real numbers and the laws of exponents.