Question

simplify the expression \\(\frac{-5x^{-6}}{(-2x^{-4})^4}\\)

Explanation:

Step1: Simplify the denominator using power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). For \((-2x^{-4})^4\), we have \((-2)^4 \times (x^{-4})^4\). Calculating \((-2)^4 = 16\) and using the power of a power rule \((a^m)^n = a^{mn}\), we get \((x^{-4})^4 = x^{-16}\). So the denominator becomes \(16x^{-16}\).

Step2: Simplify the fraction

We have \(\frac{-5x^{-6}}{16x^{-16}}\). Using the rule \(\frac{a^m}{a^n} = a^{m - n}\) for the \(x\) terms, we get \(x^{-6 - (-16)} = x^{10}\). So the fraction simplifies to \(\frac{-5}{16}x^{10}\).

Answer:

\(-\frac{5}{16}x^{10}\)