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 for \( \sqrt[3]{16} \times \sqrt[3]{625} \times \sqrt{10} \) in terms of \( y \), we first rewrite \( y \): \[ y = \sqrt[6]{100000} = 100000^{1/6} \] The number \( 100000 \) can be expressed as \( 10^5 \) since \( 100000 = 10^5 \). Thus, we have: \[ y = (10^5)^{1/6} = 10^{5/6} \] Now, let’s evaluate \( \sqrt[3]{16} \times \sqrt[3]{625} \times \sqrt{10} \). Starting with \( \sqrt[3]{16} \): \[ \sqrt[3]{16} = \sqrt[3]{2^4} = 2^{4/3} \] Next, \( \sqrt[3]{625} \): \[ \sqrt[3]{625} = \sqrt[3]{5^4} = 5^{4/3} \] Now, \( \sqrt{10} \): \[ \sqrt{10} = 10^{1/2} \] We combine these results: \[ \sqrt[3]{16} \times \sqrt[3]{625} \times \sqrt{10} = 2^{4/3} \times 5^{4/3} \times 10^{1/2} \] Since \( 10 = 2 \times 5 \), we can write \( 10^{1/2} \) as: \[ 10^{1/2} = (2 \times 5)^{1/2} = 2^{1/2} \times 5^{1/2} \] Now we can rewrite the entire expression: \[ = 2^{4/3} \times 5^{4/3} \times 2^{1/2} \times 5^{1/2} \] Combining the powers of \( 2 \) and \( 5 \): \[ = 2^{4/3 + 1/2} \times 5^{4/3 + 1/2} \] Let's find a common denominator: - For \( 4/3 \): - The denominator is 3: \( 4/3 = 8/6 \) - For \( 1/2 \): - The denominator is 2: \( 1/2 = 3/6 \) So, \[ 4/3 + 1/2 = 8/6 + 3/6 = 11/6 \] Thus, the expression becomes: \[ 2^{11/6} \times 5^{11/6} = (2 \times 5)^{11/6} = 10^{11/6} \] Now, recalling that \( y = 10^{5/6} \), we can express \( 10^{11/6} \) in terms of \( y \): \[ 10^{11/6} = 10^{5/6} \times 10^{6/6} = y \times 10 \] Thus, the final expression for \( \sqrt[3]{16} \times \sqrt[3]{625} \times \sqrt{10} \) in terms of \( y \) is: \[ y \times 10 \]