Use the product property of roots to choose the expression equivalent to $\sqrt[3]{5 x} \cdot \sqrt[3]{25 x^2}$.
A. $\sqrt[4]{125 x^3}$
B. $\sqrt[3]{125 x^3}$
C. $\sqrt[3]{30 x^2}$
D. $\sqrt[3]{30 x}$
Answer (2)
Apply the product property of roots: 3 5 x ⋅ 3 25 x 2 = 3 ( 5 x ) ( 25 x 2 ) .
Simplify the expression inside the cube root: ( 5 x ) ( 25 x 2 ) = 125 x 3 .
The expression becomes 3 125 x 3 .
The equivalent expression is 3 125 x 3 .
Explanation
Understanding the problem We are asked to find an expression equivalent to 3 5 x ⋅ 3 25 x 2 using the product property of roots. The product property of roots states that n a ⋅ n b = n ab .
Applying the product property Applying the product property, we have 3 5 x ⋅ 3 25 x 2 = 3 ( 5 x ) ( 25 x 2 )
Simplifying the expression Now, we simplify the expression inside the cube root: ( 5 x ) ( 25 x 2 ) = 5 ⋅ 25 ⋅ x ⋅ x 2 = 125 x 3
Rewriting the expression Therefore, the expression becomes 3 125 x 3
Simplifying the cube root We can further simplify this expression by taking the cube root of 125 and x 3 :
3 125 x 3 = 3 125 ⋅ 3 x 3 = 5 x However, we are asked to choose from the given options, and the simplified expression 3 125 x 3 is among the choices.
Final Answer Thus, the equivalent expression is 3 125 x 3 .
Examples
The product property of roots is useful in simplifying expressions involving radicals. For example, when calculating the volume of a rectangular prism with dimensions involving cube roots, you might use this property to combine the terms and simplify the final volume calculation. Suppose the dimensions of a rectangular prism are 3 5 x , 3 25 x 2 , and 2. The volume would be 2 ⋅ 3 5 x ⋅ 3 25 x 2 = 2 ⋅ 3 125 x 3 = 2 ⋅ 5 x = 10 x .
Using the product property of roots, we simplify 3 5 x ⋅ 3 25 x 2 to get 3 125 x 3 . Among the given options, this expression corresponds to option B. Therefore, the answer is B: 3 125 x 3 .
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