Rationalize the denominator and simplify.
$\frac{-3}{\sqrt[3]{25 a^2}}$
Assume that all variables represent positive numbers.
Answer (1)
Rewrite the denominator as 3 5 2 a 2 .
Multiply the numerator and denominator by 3 5 a to rationalize the denominator.
Simplify the expression to obtain the final answer.
The rationalized and simplified expression is 5 a − 3 3 5 a .
Explanation
Understanding the Problem We are given the expression 3 25 a 2 − 3 to rationalize and simplify. All variables represent positive numbers. Our goal is to eliminate the radical from the denominator.
Rewriting the Denominator First, we rewrite the denominator to better see what we need to multiply by to rationalize it. We have 3 25 a 2 = 3 5 2 a 2 . To make the radicand a perfect cube, we need to multiply by 3 5 a .
Multiplying by the Conjugate Next, we multiply both the numerator and the denominator by 3 5 a : 3 25 a 2 − 3 ⋅ 3 5 a 3 5 a = 3 25 a 2 ⋅ 5 a − 3 3 5 a = 3 125 a 3 − 3 3 5 a .
Simplifying the Expression Now we simplify the denominator: 3 125 a 3 − 3 3 5 a = 5 a − 3 3 5 a .
Final Answer Therefore, the rationalized and simplified expression is 5 a − 3 3 5 a .
Examples
Rationalizing the denominator is a technique used to eliminate radicals from the denominator of a fraction. This is useful in various fields, such as physics and engineering, where simplified expressions are easier to work with. For example, when calculating impedance in electrical circuits, you might encounter complex numbers with radicals in the denominator. Rationalizing the denominator makes it easier to perform further calculations and comparisons.
