What is $\frac{x^{\frac{2}{3}} y}{x^{\frac{2}{3}} y^{\frac{1}{3}}}$ in simplest exponential form?
Answer (2)
By canceling the common term x 3 2 and applying the quotient rule for exponents, the expression simplifies to y 3 2 . This process involves understanding how to simplify fractions with exponents correctly. The final answer is y 3 2 .
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Cancel the common factor: x 3 2 y 3 1 x 3 2 y = y 3 1 y .
Apply the quotient rule for exponents: y 3 1 y = y 1 − 3 1 .
Simplify the exponent: y 1 − 3 1 = y 3 2 .
The simplified expression is y 3 2 .
Explanation
Understanding the Problem We are given the expression x 3 2 y 3 1 x 3 2 y . Our goal is to simplify this expression and express it in its simplest exponential form.
Canceling Common Factors First, we can cancel the common factor x 3 2 from the numerator and the denominator: x 3 2 y 3 1 x 3 2 y = y 3 1 y .
Applying the Quotient Rule Next, we simplify the expression y 3 1 y using the quotient rule for exponents, which states that a n a m = a m − n . In our case, a = y , m = 1 , and n = 3 1 . Therefore, we have y 3 1 y = y 1 − 3 1 = y 3 3 − 3 1 = y 3 2 .
Final Simplification Thus, the simplified expression is y 3 2 .
Examples
Imagine you're adjusting a recipe that calls for unusual ingredient quantities. Simplifying exponential expressions is like converting those quantities into more manageable forms, making the recipe easier to follow and measure. For example, if a recipe requires x 3 2 cups of flour and x 3 2 y 3 1 cups of sugar, simplifying the ratio helps you understand the relative amounts of each ingredient needed. This skill is useful not only in cooking but also in various fields where proportional adjustments are necessary, such as mixing chemicals or scaling architectural designs.
