Questions: (4a^5bc^-4)^-3

Solution Steps

To simplify the expression \((4 a^{5} b c^{-4})^{-3}\), we need to apply the power of a power rule, which states that \((x^m)^n = x^{m \cdot n}\). This means we will multiply the exponents inside the parentheses by \(-3\). Additionally, we will apply the rule for negative exponents, which states that \(x^{-n} = \frac{1}{x^n}\).

We start with the expression \((4 a^{5} b c^{-4})^{-3}\). By applying the power of a power rule, we distribute the exponent \(-3\) to each factor inside the parentheses:

\[ (4)^{-3} (a^{5})^{-3} (b)^{-3} (c^{-4})^{-3} \]

Now we simplify each factor:

• \((4)^{-3} = \frac{1}{4^3} = \frac{1}{64}\)
• \((a^{5})^{-3} = a^{-15}\)
• \((b)^{-3} = b^{-3}\)
• \((c^{-4})^{-3} = c^{12}\)

Putting it all together, we have:

\[ \frac{c^{12}}{64 a^{15} b^{3}} \]

Final Answer