Question

(\frac{(4x)^{4}}{12x^{-7}})

Explanation:

Step1: Simplify the numerator

Using the power of a product rule \((ab)^n = a^n b^n\), we have \((4x)^4 = 4^4 x^4\). Calculating \(4^4\), we get \(4^4 = 256\). So the numerator becomes \(256x^4\).

Step2: Simplify the expression with negative exponents

Recall that \(a^{-n}=\frac{1}{a^n}\), so \(x^{-7}=\frac{1}{x^7}\), and dividing by \(x^{-7}\) is the same as multiplying by \(x^7\). So our expression \(\frac{256x^4}{12x^{-7}}\) becomes \(\frac{256x^4\times x^7}{12}\).

Step3: Combine like terms (exponents)

Using the product rule for exponents \(a^m\times a^n = a^{m + n}\), we add the exponents of \(x\): \(x^4\times x^7=x^{4 + 7}=x^{11}\). So now our expression is \(\frac{256x^{11}}{12}\).

Step4: Simplify the fraction

We can simplify \(\frac{256}{12}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 4. \(\frac{256\div4}{12\div4}=\frac{64}{3}\). So the simplified expression is \(\frac{64}{3}x^{11}\).

Answer:

\(\frac{64}{3}x^{11}\)