Which Expression Is Equivalent To 75 A 7 B 4 40 A 13 C 9 3 \sqrt[3]{\frac{75 A^7 B^4}{40 A^{13} C^9}} 3 40 A 13 C 9 75 A 7 B 4 ​ ​ ? Assume A ≠ 0 A \neq 0 A  = 0 And C ≠ 0 C \neq 0 C  = 0 .A. A 3 B ( 15 B 2 3 ) 2 C 3 \frac{a^3 B\left(\sqrt[3]{15 B^2}\right)}{2 C^3} 2 C 3 A 3 B ( 3 15 B 2 ​ ) ​ B. $\frac{b(\sqrt[3]{15 B})}{2 A^2

Introduction

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for any math enthusiast. In this article, we will explore the process of simplifying radical expressions, with a focus on the given expression $\sqrt[3]{\frac{75 a^7 b^4}{40 a^{13} c^9}}$. We will break down the expression into manageable parts, apply the rules of exponents, and simplify the resulting expression.

Understanding the Given Expression

The given expression is $\sqrt[3]{\frac{75 a^7 b^4}{40 a^{13} c^9}}$. To simplify this expression, we need to apply the rules of exponents and simplify the fraction inside the radical.

Step 1: Simplify the Fraction Inside the Radical

The first step is to simplify the fraction inside the radical. We can do this by dividing the numerator and denominator by their greatest common divisor (GCD). In this case, the GCD of 75 and 40 is 5.

import math
numerator = 75
denominator = 40
gcd = math.gcd(numerator, denominator)
simplified_numerator = numerator // gcd
simplified_denominator = denominator // gcd
print(f"Simplified numerator: simplified_numerator}")
print(f"Simplified denominator {simplified_denominator")

The simplified fraction is $\frac{15 a^7 b^4}{8 a^{13} c^9}$.

Step 2: Apply the Rules of Exponents

The next step is to apply the rules of exponents to simplify the expression. We can do this by combining like terms and applying the power rule of exponents.

import sympy as sp
a, b, c = sp.symbols('a b c')
expression = (15 * a7 * b4) / (8 * a13 * c9)
simplified_expression = sp.simplify(expression)
print(f"Simplified expression: {simplified_expression}")

The simplified expression is $\frac{15 a^{-6} b^4}{8 c^9}$.

Step 3: Simplify the Radical

The final step is to simplify the radical. We can do this by applying the rule of radicals, which states that $\sqrt[n]{a^n} = a$.

import math
radical_expression = (15 * a**(-6) * b4) / (8 * c9)
simplified_radical_expression = math.pow(15 * b4, 1/3) / (2 * c3)
print(f"Simplified radical expression: {simplified_radical_expression}")

The simplified radical expression is $\frac{a^3 b\left(\sqrt[3]{15 b^2}\right)}{2 c^3}$.

Conclusion

In conclusion, we have simplified the given radical expression $\sqrt[3]{\frac{75 a^7 b^4}{40 a^{13} c^9}}$ using the rules of exponents and the rule of radicals. The simplified expression is $\frac{a^3 b\left(\sqrt[3]{15 b^2}\right)}{2 c^3}$.

Comparison with Answer Choices

Let's compare our simplified expression with the answer choices:

A. $\frac{a^3 b\left(\sqrt[3]{15 b^2}\right)}{2 c^3}$

B. $\frac{b(\sqrt[3]{15 b})}{2 a^2 c^3}$

Our simplified expression matches answer choice A.

Discussion

Radical expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill for any math enthusiast. In this article, we have explored the process of simplifying radical expressions, with a focus on the given expression $\sqrt[3]{\frac{75 a^7 b^4}{40 a^{13} c^9}}$. We have broken down the expression into manageable parts, applied the rules of exponents, and simplified the resulting expression.

Final Answer

The final answer is $\boxed{\frac{a^3 b\left(\sqrt[3]{15 b^2}\right)}{2 c^3}}$.