Which expression is equivalent to $\sqrt[3]{\frac{75 a^7 b^4}{40 a^{13} c^9}}$ ? Assume $a \neq 0$ and $c \neq 0$.
A. $\frac{a^3 b\left(\sqrt[3]{15 b^2}\right)}{2 c^3}$
B. $\frac{b(\sqrt[3]{15 b})}{2 a^2 c^3}$
C. $\frac{a^3 b\left(\sqrt[3]{15 b^2}\right)}{6 c^3}$
Answer (2)
The expression 3 40 a 13 c 9 75 a 7 b 4 simplifies to 2 a 2 c 3 b 3 15 b . Therefore, the correct answer from the options given is B.
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Simplify the fraction inside the cube root: 40 a 13 c 9 75 a 7 b 4 = 8 a 6 c 9 15 b 4 .
Apply the cube root: 3 8 a 6 c 9 15 b 4 = 3 8 a 6 c 9 3 15 b 4 .
Simplify the numerator: 3 15 b 4 = b 3 15 b .
Simplify the denominator: 3 8 a 6 c 9 = 2 a 2 c 3 .
Combine the terms: The final simplified expression is 2 a 2 c 3 b ( 3 15 b ) .
Explanation
Understanding the Problem We are asked to simplify the expression 3 40 a 13 c 9 75 a 7 b 4 . Let's break this down step by step.
Simplifying the Fraction First, simplify the fraction inside the cube root: 40 a 13 c 9 75 a 7 b 4 = 8 × 5 a 13 c 9 15 × 5 a 7 b 4 = 8 a 13 − 7 c 9 15 b 4 = 8 a 6 c 9 15 b 4
Applying the Cube Root Now, apply the cube root to the simplified fraction: 3 8 a 6 c 9 15 b 4 = 3 8 a 6 c 9 3 15 b 4
Simplifying Cube Roots Next, simplify the cube roots in the numerator and the denominator separately. For the numerator: 3 15 b 4 = 3 15 b 3 ⋅ b = 3 b 3 ⋅ 3 15 b = b 3 15 b For the denominator: 3 8 a 6 c 9 = 3 2 3 ( a 2 ) 3 ( c 3 ) 3 = 2 a 2 c 3
Combining Terms Combine the simplified numerator and denominator: 2 a 2 c 3 b 3 15 b
Final Answer Therefore, the simplified expression is 2 a 2 c 3 b 3 15 b .
Examples
Imagine you are calculating the volume of a complex 3D shape, and the formula involves cube roots of fractions with variables. Simplifying such expressions, as we did here, makes the calculation much easier and helps you understand how different dimensions contribute to the overall volume. This skill is useful in fields like engineering, architecture, and computer graphics, where dealing with complex geometric shapes is common.
