Question

1. \\(\frac{(p^{-3})^{-4}}{2pp^{-3}}\\)

Explanation:

Step1: Simplify the numerator using power of a power rule

The power of a power rule states that \((a^m)^n = a^{m \times n}\). For the numerator \((p^{-3})^{-4}\), we have \(m=-3\) and \(n = - 4\), so \((p^{-3})^{-4}=p^{(-3)\times(-4)}=p^{12}\)

Step2: Simplify the denominator using product of powers rule

The product of powers rule states that \(a^m\times a^n=a^{m + n}\). For the denominator \(2pp^{-3}\), we have \(a = p\), \(m = 1\) and \(n=-3\), so \(pp^{-3}=p^{1+( - 3)}=p^{-2}\), and the denominator is \(2p^{-2}\)

Step3: Simplify the fraction

Now we have \(\frac{p^{12}}{2p^{-2}}\). Using the quotient of powers rule \(\frac{a^m}{a^n}=a^{m - n}\), we get \(\frac{1}{2}p^{12-( - 2)}=\frac{1}{2}p^{14}\)

Answer:

\(\frac{1}{2}p^{14}\)