Question

Using the exponent properties, which of the following expressions is equivalent to (√64)^2/3⋅64^0
?(1 point)
Responses

4
4

64^−1/3
64 superscript negative Start Fraction 1 over 3 End Fraction baseline

1
1

1/4
Start Fraction 1 over 4 End Fraction

Answer 1

To simplify the expression \((\sqrt{64})^{\frac{2}{3}} \cdot 64^0\), we can break it down step by step using exponent properties.

1. **Simplify \(64^0\)**:
\[
64^0 = 1 \quad \text{(any number to the power of 0 is 1)}
\]

2. **Simplify \(\sqrt{64}\)**:
\[
\sqrt{64} = 64^{\frac{1}{2}} \quad \text{(since square root is the same as raising to the power of } \frac{1}{2}\text{)}
\]

3. **Now substitute \(\sqrt{64}\) in the original expression**:
\[
(\sqrt{64})^{\frac{2}{3}} = (64^{\frac{1}{2}})^{\frac{2}{3}}
\]

4. **Apply the power of a power property \((a^m)^n = a^{m \cdot n}\)**:
\[
(64^{\frac{1}{2}})^{\frac{2}{3}} = 64^{\frac{1}{2} \cdot \frac{2}{3}} = 64^{\frac{2}{6}} = 64^{\frac{1}{3}}
\]

5. **Now multiply the results**:
\[
64^{\frac{1}{3}} \cdot 1 = 64^{\frac{1}{3}}
\]

Now we need to compare \(64^{\frac{1}{3}}\) with the provided options. We can write \(64\) as \(64^{\frac{1}{3}}\):

- Rewrite \(64\) in terms of its base: \(64 = 4^3\), so:
\[
64^{\frac{1}{3}} = (4^3)^{\frac{1}{3}} = 4^{3 \cdot \frac{1}{3}} = 4^1 = 4
\]

Thus, the expression \((\sqrt{64})^{\frac{2}{3}} \cdot 64^0\) simplifies to:

\[
4
\]

So the final answer is: **4**.