Simplify ($\sqrt[3]{64a^3})^{-1}$

1. Analyze the Question

We are asked to simplify the expression $(\sqrt[3]{64a^3})^{-1}$. This involves two main operations: finding the cube root of the terms inside the bracket and then applying the negative exponent rule.

2. Explain the Concept

• Cube Root ($\sqrt[3]{x}$): This is equivalent to raising a number to the power of $1/3$. So, $\sqrt[3]{64a^3} = (64a^3)^{1/3}$.
• Power of a Product Rule: $(xy)^n = x^n \cdot y^n$. We apply this to both the constant (64) and the variable ($a^3$).
• Negative Exponent Rule: $x^{-n} = \frac{1}{x^n}$. This rule tells us that a negative exponent indicates the reciprocal of the base.

3. Show the Solution

Step 1: Simplify the expression inside the bracket. We need to find the cube root of $64a^3$:

• Since $4 \times 4 \times 4 = 64$, the cube root of 64 is 4.
• The cube root of $a^3$ is simply $a$ (because $a \times a \times a = a^3$).

So, the expression inside the bracket simplifies to $4a$.

Step 2: Apply the negative exponent. Now we substitute this back into the original expression:

Using the negative exponent rule ($x^{-1} = \frac{1}{x}$):

4. Identify the Answer

Comparing our result to the given options, we can see that our solution matches option D.

5. Tip

When dealing with cube roots of variables, remember that $\sqrt[3]{x^n} = x^{n/3}$. If the power is a multiple of 3, it will always simplify perfectly to a whole number power. For constants like 64, always look for the number that, when multiplied by itself three times, gives you the value inside the root!