Question

\frac{(4w^{2})^{3}}{20w^{5}}

Explanation:

Step1: Simplify the numerator using power of a product rule

The power of a product rule states that \((ab)^n = a^n b^n\). So, \((4w^{2})^{3}=4^{3}\times(w^{2})^{3}\). Calculating \(4^{3}=64\) and using the power of a power rule \((a^{m})^{n}=a^{mn}\), we get \((w^{2})^{3}=w^{2\times3}=w^{6}\). So the numerator becomes \(64w^{6}\).

Step2: Rewrite the fraction with the simplified numerator

Now our fraction is \(\frac{64w^{6}}{20w^{5}}\).

Step3: Simplify the coefficients and the variables separately

For the coefficients, we simplify \(\frac{64}{20}\) by dividing both numerator and denominator by their greatest common divisor, which is 4. So \(\frac{64\div4}{20\div4}=\frac{16}{5}\). For the variables, we use the quotient rule for exponents \(a^{m}\div a^{n}=a^{m - n}\), so \(w^{6}\div w^{5}=w^{6-5}=w^{1}=w\).

Step4: Combine the simplified coefficients and variables

Multiplying the simplified coefficient and variable together, we get \(\frac{16}{5}w\) or \(\frac{16w}{5}\).

Answer:

\(\frac{16w}{5}\)