Which expression is equivalent to $\frac{\sqrt[7]{x^2}}{\sqrt[5]{y^3}}$ ? Assume $y \neq 0$.
A. $\left(x^{\frac{2}{7}}\right)\left(y^{-\frac{3}{5}}\right)$
B. $\left(x^{\frac{2}{7}}\right)\left(y^{\frac{5}{3}}\right)$
C. $\left(x^{\frac{2}{7}}\right)\left(y^{\frac{3}{5}}\right)$
D. $\left(x^{\frac{7}{2}}\right)\left(y^{-\frac{5}{3}}\right)$
Answer (2)
The equivalent expression for 5 y 3 7 x 2 is ( x 7 2 ) ( y − 5 3 ) , which is option A. This was determined by converting radicals to exponents and applying negative exponent rules. Therefore, the correct answer is option A.
;
Convert the radicals to exponents: 7 x 2 = x 7 2 and 5 y 3 = y 5 3 .
Rewrite the expression: 5 y 3 7 x 2 = y 5 3 x 7 2 .
Use the property of negative exponents: y 5 3 1 = y − 5 3 .
The equivalent expression is ( x 7 2 ) ( y − 5 3 ) .
Explanation
Understanding the Problem We are asked to find an expression equivalent to 5 y 3 7 x 2 . This involves understanding how to convert radicals to exponents and how to handle division with exponents.
Converting Radicals to Exponents First, let's convert the radicals in the numerator and the denominator to exponential form. Recall that n a m = a n m . Therefore, we have: 7 x 2 = x 7 2 5 y 3 = y 5 3
Rewriting the Expression Now, we can rewrite the original expression as: 5 y 3 7 x 2 = y 5 3 x 7 2
Using Negative Exponents To express this without a fraction, we can use the property that a n 1 = a − n . Applying this to the denominator, we get: y 5 3 x 7 2 = x 7 2 y − 5 3
Final Answer Therefore, the equivalent expression is x 7 2 y − 5 3 , which matches the first option.
Examples
Understanding how to manipulate radicals and exponents is crucial in many areas of mathematics and physics. For example, when dealing with physical quantities that scale with certain powers (like the area of a circle scaling with the square of the radius, A = π r 2 ), being able to convert between radical and exponential forms allows for easier manipulation and calculation. In finance, understanding exponential growth (or decay) is essential for calculating compound interest or depreciation. For instance, if an investment grows at a rate proportional to its current value, the value at time t can be expressed as V ( t ) = V 0 e k t , where V 0 is the initial investment and k is the growth rate. Converting between exponential and radical forms can help in comparing different investment options or understanding the time it takes for an investment to reach a certain value.
