Question
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Tutor-Verified Answer
Answer
The final answer is 8.
Solution
1. Recognize that \(1,024 = 2^{10}\) and \(32 = 2^5\).
Therefore,
\[
1,024 \cdot 32 = 2^{10} \cdot 2^5 = 2^{10+5} = 2^{15}.
\]
2. Given the expression \(\left(2^{15}\right)^{\frac{1}{5}}\), apply the power rule:
\[
\left(2^{15}\right)^{\frac{1}{5}} = 2^{\frac{15}{5}} = 2^3.
\]
3. Finally, compute:
\[
2^3 = 8.
\]
Thus, the final answer is \(8\).
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Simplify this solution Beyond the Answer
Let's break this down! First, \(1,024\) is actually \(2^{10}\) and \(32\) is \(2^5\). So, when you multiply them together: \[ 1,024 \cdot 32 = 2^{10} \cdot 2^5 = 2^{15} \] Now, taking the fifth root: \[ (2^{15})^{\frac{1}{5}} = 2^{15/5} = 2^3 = 8 \] So, \( (1,024 \cdot 32)^{\frac{1}{5}} = 8\). Isn't it cool how numbers can transform into something elegant and simple like that?
