Question

Use the properties of exponents to generate an equivalent expression for 8√32√
8
3
2
.(1 point)
2^?

Answer 1

To solve for the equivalent expression of \( 8\sqrt{32}\sqrt[3]{2} \) using the properties of exponents, we need to express all the numbers as powers of 2.

1. **Convert 8 to a power of 2:**
\[
8 = 2^3
\]

2. **Convert 32 to a power of 2:**
\[
32 = 2^5
\]
Therefore,
\[
\sqrt{32} = \sqrt{2^5} = 2^{5/2}
\]

3. **Convert \( \sqrt[3]{2} \) to an exponent:**
\[
\sqrt[3]{2} = 2^{1/3}
\]

4. **Now combine all these expressions together:**
\[
8\sqrt{32}\sqrt[3]{2} = 2^3 \cdot 2^{5/2} \cdot 2^{1/3}
\]

5. **Use the properties of exponents to combine:**
\[
= 2^{3 + \frac{5}{2} + \frac{1}{3}}
\]

6. **Find a common denominator to combine the exponents. The least common multiple of 1, 2, and 3 is 6. Convert each term:**
- For 3: \( 3 = \frac{18}{6} \)
- For \( \frac{5}{2} \): \( \frac{5}{2} = \frac{15}{6} \)
- For \( \frac{1}{3} \): \( \frac{1}{3} = \frac{2}{6} \)

7. **Now combine them:**
\[
3 + \frac{5}{2} + \frac{1}{3} = \frac{18}{6} + \frac{15}{6} + \frac{2}{6} = \frac{35}{6}
\]

8. **Thus, the equivalent expression is:**
\[
8\sqrt{32}\sqrt[3]{2} = 2^{\frac{35}{6}}
\]

So, in the form \( 2^? \), we have:
\[
2^{\frac{35}{6}}
\]