$\frac{2 x}{\sqrt[4]{125 x^3}}$
Answer (1)
Rewrite the expression with fractional exponents.
Simplify the exponents in the numerator and denominator.
Combine like terms.
Rationalize the denominator (optional).
The simplified expression is 5 2 4 5 x .
Explanation
Understanding the Problem We are given the expression 4 125 x 3 2 x and we want to simplify it.
Rewriting with Fractional Exponents First, let's rewrite the expression using fractional exponents. Recall that n a = a n 1 . So, we have 4 125 x 3 2 x = ( 125 x 3 ) 4 1 2 x
Distributing the Exponent Next, distribute the exponent in the denominator: ( 125 x 3 ) 4 1 2 x = 12 5 4 1 ( x 3 ) 4 1 2 x
Simplifying Exponents Now, simplify the exponents. Recall that ( a m ) n = a mn . So, ( x 3 ) 4 1 = x 4 3 . Thus, 12 5 4 1 x 4 3 2 x
Rewriting 125 Rewrite 125 as 5 3 :
( 5 3 ) 4 1 x 4 3 2 x
Simplifying the Exponent Simplify the exponent: ( 5 3 ) 4 1 = 5 4 3 . So, 5 4 3 x 4 3 2 x
Combining x Terms Now, combine the terms with x . We have x 4 3 x = x 1 − 4 3 = x 4 1 . So, 5 4 3 x 4 3 2 x = 5 4 3 2 x 4 1
Rewriting with Radicals Rewrite the expression using radicals: 5 4 3 2 x 4 1 = 4 5 3 2 4 x = 4 125 2 4 x
Rationalizing the Denominator Finally, we can rationalize the denominator (optional). Multiply the numerator and denominator by 4 5 :
4 125 2 4 x ⋅ 4 5 4 5 = 4 125 ⋅ 5 2 4 5 x = 4 625 2 4 5 x = 5 2 4 5 x
Final Answer Therefore, the simplified expression is 5 2 4 5 x .
Examples
Imagine you are calculating the flow rate of a fluid through a pipe, and the formula involves simplifying radicals like the one in this problem. Simplifying the expression makes it easier to calculate the flow rate for different pipe sizes and fluid viscosities. This type of simplification is also useful in electrical engineering when dealing with impedance calculations involving complex numbers and fractional exponents. By simplifying such expressions, engineers can more easily analyze and design circuits.
