If \( y=\sqrt[6]{100000} \), WITHOUT USING A CALCULATOR, determine the value \( \sqrt[3]{16} \times \sqrt[3]{625} \times \sqrt{10} \) in terms of \( y \)

To solve the expression \( \sqrt[3]{16} \times \sqrt[3]{625} \times \sqrt{10} \) in terms of \( y = \sqrt[6]{100000} \), we first calculate each part. 1. Calculate \( \sqrt[3]{16} \): \[ \sqrt[3]{16} = \sqrt[3]{2^4} = 2^{4/3} \] 2. Calculate \( \sqrt[3]{625} \): \[ 625 = 5^4 \quad \Rightarrow \quad \sqrt[3]{625} = \sqrt[3]{5^4} = 5^{4/3} \] 3. Calculate \( \sqrt{10} \): \[ \sqrt{10} = 10^{1/2} = (2 \cdot 5)^{1/2} = 2^{1/2} \cdot 5^{1/2} \] Now combine all parts: \[ \sqrt[3]{16} \times \sqrt[3]{625} \times \sqrt{10} = 2^{4/3} \times 5^{4/3} \times (2^{1/2} \cdot 5^{1/2}) \] Combining the terms for \( 2 \) and \( 5 \): \[ = 2^{4/3 + 1/2} \cdot 5^{4/3 + 1/2} \] We need a common denominator to combine the exponents. The least common multiple of \( 3 \) and \( 2 \) is \( 6 \): - For \( 4/3 \): \( \frac{4}{3} = \frac{8}{6} \) - For \( 1/2 \): \( \frac{1}{2} = \frac{3}{6} \) So: \[ 4/3 + 1/2 = \frac{8}{6} + \frac{3}{6} = \frac{11}{6} \] Thus, we have: \[ 2^{11/6} \cdot 5^{11/6} = (2 \cdot 5)^{11/6} = 10^{11/6} \] Now, express \( y \): \[ y = \sqrt[6]{100000} = \sqrt[6]{10^5} = 10^{5/6} \] Now we relate \( 10^{11/6} \) to \( y \): \[ 10^{11/6} = 10^{(6/6 + 5/6)} = 10^{1} \cdot 10^{5/6} = 10 \cdot y \] Thus, we find: \[ \sqrt[3]{16} \times \sqrt[3]{625} \times \sqrt{10} = 10y \] So the final answer is: \[ \boxed{10y} \]