How do you simplify the square root of 3 multiplied by the fifth root of 3?

To simplify the expression √3 × &radic[5]{3}, we can rewrite the roots using fractional exponents. The square root of 3 can be expressed as 3^1/2 and the fifth root of 3 can be expressed as 3^1/5. Thus, the original expression becomes:

3^1/2 × 3^1/5

When multiplying exponential terms with the same base, we can add the exponents. Therefore, we have:

3^{1/2 + 1/5}

To combine the exponents, we first need to find a common denominator for 2 and 5. The least common multiple (LCM) of 2 and 5 is 10. So we convert the fractions:

• 1/2 = 5/10
• 1/5 = 2/10

This gives us:

1/2 + 1/5 = 5/10 + 2/10 = 7/10

Now we can rewrite our expression:

3^7/10

This means that the simplified form of the square root of 3 multiplied by the fifth root of 3 is:

&radic[10]{3⁷}