$\frac{16 x}{\sqrt[3]{36 x^2}}$
Answer (2)
Rewrite the expression with fractional exponents.
Simplify the denominator by distributing the exponent.
Combine the x terms.
Rationalize the denominator.
The simplified expression is 3 8 3 6 x .
Explanation
Problem Analysis We are given the expression 3 36 x 2 16 x and we want to simplify it.
Fractional Exponent First, rewrite the cube root in the denominator using a fractional exponent: ( 36 x 2 ) 1/3 16 x .
Distributing the Exponent Next, distribute the exponent in the denominator: 3 6 1/3 ( x 2 ) 1/3 16 x = 3 6 1/3 x 2/3 16 x .
Rewriting x Rewrite x in the numerator as x 3/3 : 3 6 1/3 x 2/3 16 x 3/3 .
Combining x Terms Simplify the expression by combining the x terms: 3 6 1/3 x 2/3 16 x 3/3 = 3 6 1/3 16 x ( 3/3 − 2/3 ) = 3 6 1/3 16 x 1/3 .
Simplifying the Constant Rewrite 3 6 1/3 as ( 2 2 3 2 ) 1/3 = 2 2/3 3 2/3 . So the expression becomes 2 2/3 3 2/3 16 x 1/3 .
Rationalizing the Denominator To rationalize the denominator, we want to multiply both the numerator and the denominator by a term that will eliminate the fractional exponents in the denominator. We can multiply by 2 1/3 3 1/3 : 2 2/3 3 2/3 16 x 1/3 ⋅ 2 1/3 3 1/3 2 1/3 3 1/3 = 2 2/3 + 1/3 3 2/3 + 1/3 16 ⋅ 2 1/3 3 1/3 x 1/3 = 2 1 3 1 16 ⋅ 2 1/3 3 1/3 x 1/3 = 6 16 ⋅ 2 1/3 3 1/3 x 1/3 = 3 8 2 1/3 3 1/3 x 1/3 .
Final Simplification We can rewrite this as 3 8 ( 2 ⋅ 3 ) 1/3 x 1/3 = 3 8 ( 6 ) 1/3 x 1/3 = 3 8 3 6 x .
Final Answer Thus, the simplified expression is 3 8 3 6 x .
Examples
Imagine you are calculating the volume of a special type of crystal that grows in a cubic manner, but its dimensions are not perfectly aligned with the axes. Simplifying expressions like the one above can help in determining the crystal's growth rate or volume more efficiently. For instance, if 'x' represents a growth factor, simplifying the expression allows for easier computation and analysis of the crystal's properties. This type of simplification is crucial in material science and crystallography for understanding complex structures and their properties.
To simplify 3 36 x 2 16 x , we rewrite it with fractional exponents, combine like terms, and rationalize the denominator. The final simplified expression is 3 8 3 6 x .
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