Question

Using the exponent properties, which of the following expressions is equivalent to (64−−√)23⋅640
?(1 point)
Responses

4
4

14
Start Fraction 1 over 4 End Fraction

1
1

64−13

Answer 1

To simplify the expression \( (64^{\frac{1}{2}})^{23} \cdot 64^0 \), we can use the properties of exponents.

1. First, simplify \( 64^{\frac{1}{2}} \):
\[
64^{\frac{1}{2}} = \sqrt{64} = 8
\]

2. Now, substitute this back into the expression:
\[
(64^{\frac{1}{2}})^{23} = 8^{23}
\]

3. The term \( 64^0 \) equals 1, since any number raised to the power of 0 is 1:
\[
64^0 = 1
\]

4. Now we combine the two parts:
\[
8^{23} \cdot 1 = 8^{23}
\]

5. To express \( 8 \) in terms of \( 64 \), we write \( 8 \) as \( 64^{\frac{1}{3}} \):
\[
8^{23} = (64^{\frac{1}{3}})^{23} = 64^{\frac{23}{3}}
\]

Now let's look at the original expression:
- \( (64^{\frac{1}{2}})^{23} \cdot 64^0 \) simplifies to \( 64^{\frac{23}{3}} \).

The expression \( 64^{-\frac{1}{3}} \) indicates that the exponent is negative, so we interpret as:
- \( 64^{-13} \) can show that it is equal to \( \frac{1}{64^{13}} \).

Now let's review the original options to find equivalents:

The closest interpretation to our derived exponent \( 64^{\frac{23}{3}} \) aligns with \( 64^{-\frac{1}{3}} \).

So, looking at the response choices you provided, the expression equivalent to \( (64^{\frac{1}{2}})^{23} \cdot 64^0 \) is:

**None of the response options** seem correctly simplified or equal to the derived expression. Please check your options or provide additional clarifications.

Alternatively, the correct calculation yields:
**Option: \( 64^{\frac{23}{3}} \)** which is equivalent to \( 4^{23} \) if formulated further down using the spectrum of exponent reductions.