To simplify the expression \((256\sqrt{10})^5 / 512^{1/3}\), we can follow these steps using properties of exponents and radicals.
Step 1: First simplify each part of the expression.
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Calculate \(512^{1/3}\): \[ 512 = 2^9 \quad (\text{since } 512 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2) \] Therefore, \[ 512^{1/3} = (2^9)^{1/3} = 2^{9/3} = 2^3 = 8 \]
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Calculate \((256\sqrt{10})^5\): We recognize that \(256 = 2^8\); thus, \[ 256\sqrt{10} = 2^8 \cdot 10^{1/2} \] Now raising this to the 5th power, \[ (256\sqrt{10})^5 = (2^8 \cdot 10^{1/2})^5 = (2^8)^5 \cdot (10^{1/2})^5 \] Calculating these, \[ (2^8)^5 = 2^{40}, \quad (10^{1/2})^5 = 10^{5/2} \] Therefore, \[ (256\sqrt{10})^5 = 2^{40} \cdot 10^{5/2} \]
Step 2: Now put these together in the original expression.
So far we have: \[ \frac{(256\sqrt{10})^5}{512^{1/3}} = \frac{2^{40} \cdot 10^{5/2}}{8} = \frac{2^{40} \cdot 10^{5/2}}{2^3} \]
Step 3: Simplify the fraction:
Using the properties of exponents, \[ \frac{2^{40}}{2^3} = 2^{40 - 3} = 2^{37} \] Thus, we simplify the expression to: \[ 2^{37} \cdot 10^{5/2} \]
Final Expression:
Now, we have: \[ (256\sqrt{10})^5 / 512^{1/3} = 2^{37} \cdot 10^{5/2} \]
So the final equivalent expression in simplest form is: \[ \boxed{2^{37} \cdot 10^{5/2}} \]
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