Write \( \sqrt{q^{3}} \sqrt[3]{q^{8}} \) as a single radical using the smallest possible root. Answer Attempt 1 out of 2 \( \sqrt[\square]{\square} \)

To combine \( \sqrt{q^{3}} \) and \( \sqrt[3]{q^{8}} \) into a single radical, we first convert both expressions into exponents: - \( \sqrt{q^{3}} = q^{3/2} \) - \( \sqrt[3]{q^{8}} = q^{8/3} \) Now, we can add the exponents together, using a common denominator: - The least common multiple of 2 and 3 is 6. - Convert \( q^{3/2} \) to have a denominator of 6: \( q^{3/2} = q^{9/6} \). - Convert \( q^{8/3} \) to have a denominator of 6: \( q^{8/3} = q^{16/6} \). Now we can combine them: \[ q^{9/6} \cdot q^{16/6} = q^{(9 + 16)/6} = q^{25/6} \] Now, convert \( q^{25/6} \) back into a radical form: \[ q^{25/6} = \sqrt[6]{q^{25}} \] Thus, the final answer in the requested format is: \[ \sqrt[6]{q^{25}} \]