Question

express the following fraction in simplest form using only positive exponents.
\frac{(4p^4)^4}{12p^7}
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time remaining: 120 mins

Explanation:

Step1: Simplify the numerator's power

First, we simplify \((4p^4)^4\) using the power - of - a - product rule \((ab)^n=a^n\times b^n\) and the power - of - a - power rule \((a^m)^n=a^{m\times n}\).
For \((4p^4)^4\), we have \(4^4\times(p^4)^4\).
Calculate \(4^4 = 256\) and \((p^4)^4=p^{4\times4}=p^{16}\). So \((4p^4)^4 = 256p^{16}\).
The original fraction is \(\frac{(4p^4)^4}{12p^7}=\frac{256p^{16}}{12p^7}\).

Step2: Simplify the coefficient and the variable

Simplify the coefficient \(\frac{256}{12}\). We can divide both the numerator and the denominator by their greatest common divisor, which is 4.
\(\frac{256\div4}{12\div4}=\frac{64}{3}\).
For the variable part, we use the quotient rule of exponents \(\frac{a^m}{a^n}=a^{m - n}\) (\(a
eq0\), \(m>n\)). Here, for \(\frac{p^{16}}{p^7}\), we have \(p^{16 - 7}=p^9\).

Step3: Combine the results

Multiply the simplified coefficient and the simplified variable part together. So \(\frac{256p^{16}}{12p^7}=\frac{64}{3}p^9\).

Answer:

\(\frac{64}{3}p^{9}\)