Question

simplify the expression. assume that variables represent non - zero real numbers.
$-4r^{-5}(r^{4})^{4}$
$-4r^{-5}(r^{4})^{4}=\frac{4}{r^{17}}$ (simplify your answer. type in exponential notation. type positive exponents.)

Explanation:

Step1: Apply exponent power rule

First, use the power - of - a - power rule \((a^{m})^{n}=a^{m\times n}\) on \((r^{4})^{4}\). So \((r^{4})^{4}=r^{4\times4}=r^{16}\). The expression becomes \(- 4r^{-5}\times r^{16}\).

Step2: Apply exponent product rule

Then, use the product rule of exponents \(a^{m}\times a^{n}=a^{m + n}\) for the terms with base \(r\). For \(r^{-5}\times r^{16}\), we have \(r^{-5 + 16}=r^{11}\). And the coefficient is \(-4\). So the simplified expression is \(-4r^{11}\). Wait, there seems to be a mistake in the given right - hand side of the equation in the original problem. Let's re - do the calculation:

Starting with \(-4r^{-5}(r^{4})^{4}\)

Step 1: Simplify \((r^{4})^{4}\) using \((a^{m})^{n}=a^{mn}\), we get \(r^{4\times4}=r^{16}\). So the expression is \(-4r^{-5}\cdot r^{16}\)

Step 2: Simplify \(r^{-5}\cdot r^{16}\) using \(a^{m}\cdot a^{n}=a^{m + n}\), we have \(r^{-5+16}=r^{11}\)

Step 3: Multiply the coefficient \(-4\) with \(r^{11}\), we get \(-4r^{11}\)

Answer:

\(-4r^{11}\)