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)
Simplify the fraction inside the cube root: 40 75 = 8 15 .
Apply the quotient rule of exponents: a 13 a 7 = a − 6 .
Take the cube root of the simplified expression: 3 8 a 6 c 9 15 b 4 = 3 8 a 6 c 9 3 15 b 4 .
Simplify the cube root expression: 3 8 a 6 c 9 3 15 b 4 = 2 a 2 c 3 b ( 3 15 b ) .
2 a 2 c 3 b ( 3 15 b )
Explanation
Understanding the Problem We are given the expression 3 40 a 13 c 9 75 a 7 b 4 and we want to simplify it. We assume that a = 0 and c = 0 .
Simplifying the Fraction First, simplify the fraction 40 75 by dividing both the numerator and the denominator by their greatest common divisor, which is 5. This gives us 40 75 = 8 15 . So, the expression becomes 3 8 a 13 c 9 15 a 7 b 4 .
Applying the Quotient Rule of Exponents Next, we simplify the expression by using the quotient rule of exponents, which states that a n a m = a m − n . Applying this rule to the powers of a , we have a 13 a 7 = a 7 − 13 = a − 6 . Thus, the expression becomes 3 8 a 6 c 9 15 b 4 .
Taking the Cube Root Now, we take the cube root of the expression. We can rewrite the expression as 3 8 a 6 c 9 3 15 b 4 . We know that 3 8 = 2 , 3 a 6 = a 2 , and 3 c 9 = c 3 . Also, we can rewrite b 4 as b 3 ⋅ b , so 3 b 4 = 3 b 3 ⋅ b = b 3 b . Therefore, the expression becomes 2 a 2 c 3 b 3 15 b .
Finding the Equivalent Expression Comparing the simplified expression 2 a 2 c 3 b 3 15 b with the given options, we see that it matches the second option: 2 a 2 c 3 b ( 3 15 b ) .
Final Answer Therefore, the expression equivalent to 3 40 a 13 c 9 75 a 7 b 4 is 2 a 2 c 3 b ( 3 15 b ) .
Examples
Imagine you are designing a container to hold a certain volume of liquid. The formula 3 40 a 13 c 9 75 a 7 b 4 might represent a complex relationship between the dimensions of the container (a, b, c) and the volume it can hold. Simplifying this expression to 2 a 2 c 3 b ( 3 15 b ) allows you to easily understand how changes in the dimensions affect the volume, making the design process more efficient. This type of simplification is crucial in engineering and design, where complex formulas are often used to model real-world scenarios.
The expression 3 40 a 13 c 9 75 a 7 b 4 simplifies to 2 a 2 c 3 b ( 3 15 b ) , which matches Option B. Thus, the equivalent expression is Option B.
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