Write $\sqrt{x^3} \sqrt[3]{y}$ using fractional exponents.
Answer (2)
To express x 3 3 y in fractional exponents, rewrite x 3 as x 2 3 and 3 y as y 3 1 . Then, multiply the two results to get x 2 3 y 3 1 .
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Rewrite the radicals as fractional exponents: x 3 = ( x 3 ) 2 1 and 3 y = y 3 1 .
Simplify the first term using the power of a power rule: ( x 3 ) 2 1 = x 2 3 .
Multiply the simplified terms: x 2 3 y 3 1 .
The expression x 3 3 y written with fractional exponents is x 2 3 y 3 1 .
Explanation
Understanding the problem We are asked to express x 3 3 y using fractional exponents.
Recalling the properties of radicals and exponents Recall that a radical can be expressed as a fractional exponent. Specifically, n a = a n 1 .
Rewriting the first term First, let's rewrite x 3 using a fractional exponent. We have x 3 = ( x 3 ) 2 1 . Using the power of a power rule, ( a m ) n = a mn , we get ( x 3 ) 2 1 = x 3 ⋅ 2 1 = x 2 3 .
Rewriting the second term Next, let's rewrite 3 y using a fractional exponent. We have 3 y = y 3 1 .
Multiplying the terms Now, we multiply the two terms: x 2 3 ⋅ y 3 1 .
Final Answer Therefore, x 3 3 y = x 2 3 y 3 1 .
Examples
Fractional exponents are useful in various fields, such as physics and engineering, when dealing with equations involving roots and powers. For example, when calculating the period of a pendulum, the formula involves a square root, which can be expressed as a fractional exponent. Similarly, in electrical engineering, impedance calculations often involve square roots and cube roots, which can be simplified using fractional exponents. Understanding fractional exponents allows for easier manipulation and simplification of these equations, leading to more efficient problem-solving.
