Question
Use the properties of exponents and radicals to generate an equivalent expression for (256^1/10)^5/512^1/3
Answers
GPT-4o mini
To simplify the expression \((256^{1/10})^{5}/(512^{1/3})\), we can start by simplifying the components separately.
1. **Simplifying \(256^{1/10}\):**
\[
256 = 2^8 \implies 256^{1/10} = (2^8)^{1/10} = 2^{8/10} = 2^{4/5}
\]
2. **Raising \(256^{1/10}\) to the power of 5:**
\[
(256^{1/10})^{5} = (2^{4/5})^{5} = 2^{(4/5) \cdot 5} = 2^4 = 16
\]
3. **Simplifying \(512^{1/3}\):**
\[
512 = 2^9 \implies 512^{1/3} = (2^9)^{1/3} = 2^{9/3} = 2^3 = 8
\]
4. **Putting it all together:**
Now we can rewrite the original expression:
\[
\frac{(256^{1/10})^5}{512^{1/3}} = \frac{16}{8} = 2
\]
Thus, the equivalent expression for \((256^{1/10})^{5}/(512^{1/3})\) is \(\boxed{2}\).
1. **Simplifying \(256^{1/10}\):**
\[
256 = 2^8 \implies 256^{1/10} = (2^8)^{1/10} = 2^{8/10} = 2^{4/5}
\]
2. **Raising \(256^{1/10}\) to the power of 5:**
\[
(256^{1/10})^{5} = (2^{4/5})^{5} = 2^{(4/5) \cdot 5} = 2^4 = 16
\]
3. **Simplifying \(512^{1/3}\):**
\[
512 = 2^9 \implies 512^{1/3} = (2^9)^{1/3} = 2^{9/3} = 2^3 = 8
\]
4. **Putting it all together:**
Now we can rewrite the original expression:
\[
\frac{(256^{1/10})^5}{512^{1/3}} = \frac{16}{8} = 2
\]
Thus, the equivalent expression for \((256^{1/10})^{5}/(512^{1/3})\) is \(\boxed{2}\).