The expression $\sqrt[3]{2^5} \cdot \sqrt{2}$ is equivalent to
Answer (1)
Rewrite radicals as fractional exponents: 3 2 5 = 2 5/3 and 2 = 2 1/2 .
Add the exponents when multiplying: 2 5/3 ⋅ 2 1/2 = 2 ( 5/3 + 1/2 ) = 2 13/6 .
Rewrite the fractional exponent as a radical: 2 13/6 = 6 2 13 .
Simplify the radical: 6 2 13 = 4 6 2 .
Explanation
Understanding the Problem We are given the expression 3 2 5 ⋅ 2 and we want to find an equivalent expression. This involves simplifying radicals and using properties of exponents.
Converting Radicals to Exponents First, we rewrite the radicals as fractional exponents. Recall that n a m = a n m . Therefore, 3 2 5 = 2 3 5 and 2 = 2 2 1 .
Adding the Exponents Now we multiply the terms by adding the exponents: 2 3 5 ⋅ 2 2 1 = 2 ( 3 5 + 2 1 ) . To add the fractions in the exponent, we need a common denominator, which is 6. So, 3 5 + 2 1 = 6 10 + 6 3 = 6 13 .
Converting Back to Radical Form Thus, the expression simplifies to 2 6 13 . We can rewrite this fractional exponent as a radical: 2 6 13 = 6 2 13 .
Simplifying the Radical To simplify the radical, we factor out the largest possible power of 2 that is a multiple of 6. We have 2 13 = 2 12 ⋅ 2 = ( 2 2 ) 6 ⋅ 2 = 4 6 ⋅ 2 . Therefore, 6 2 13 = 6 4 6 ⋅ 2 = 6 4 6 ⋅ 6 2 = 4 6 2 .
Examples
Understanding radical expressions is crucial in fields like physics and engineering. For example, when calculating the period of a pendulum, the formula involves a square root. Simplifying such expressions allows for easier calculations and a better understanding of the relationships between different variables. In music, the frequency of a string is related to the square root of its tension. Simplifying radical expressions helps musicians and instrument designers understand how changes in tension affect the pitch of a string.
