Which expression is equivalent to $\frac{\sqrt[4]{6}}{\sqrt[3]{2}}$ ?
A. $\frac{\sqrt[12]{27}}{2}$
B. $\frac{\sqrt[4]{24}}{2}$
C. $\frac{\sqrt[12]{55296}}{2}$
D. $\frac{\sqrt[12]{177147}}{3}$
Answer (2)
The expression 3 2 4 6 is equivalent to 2 12 55296 . Therefore, the correct answer is option C. We derived this by using fractional exponents and a common denominator for the exponents.
;
Rewrite the given expression using fractional exponents: 3 2 4 6 = 2 1/3 6 1/4 .
Rewrite the expression with a common denominator in the exponents: 2 1/3 6 1/4 = 2 4/12 6 3/12 .
Rewrite the expression using radicals with index 12 and combine them: 12 2 4 12 6 3 = 12 16 216 = 12 2 27 .
Rewrite the expression to match the form of the answer choices: 12 2 27 = 2 12 55296 .
2 12 55296
Explanation
Understanding the Problem We are given the expression 3 2 4 6 and four possible equivalent expressions. Our goal is to determine which of the four expressions is equivalent to the given expression.
Fractional Exponents Let's rewrite the given expression using fractional exponents: 3 2 4 6 = 2 1/3 6 1/4 .
Common Denominator To compare the given expression with the answer choices, we want to rewrite the expression with a common denominator in the exponents. The least common multiple of 3 and 4 is 12, so we want to express both exponents with a denominator of 12: 2 1/3 6 1/4 = 2 4/12 6 3/12 .
Radicals with Index 12 Now, rewrite the expression using radicals with index 12: 2 4/12 6 3/12 = 12 2 4 12 6 3 = 12 16 12 216 .
Single Radical Combine the radicals into a single radical: 12 16 12 216 = 12 16 216 = 12 2 27 .
Matching the Form Rewrite the expression to match the form of the answer choices. We want to get the expression into the form y 12 x for some integer x and y. To do this, multiply the numerator and denominator of the fraction inside the radical by 2 11 : 12 2 27 = 12 2 ⋅ 2 11 27 ⋅ 2 11 = 12 2 12 27 ⋅ 2 11 = 2 12 27 ⋅ 2 11 = 2 12 27 ⋅ 2048 = 2 12 55296 .
Final Answer Comparing the simplified expression with the answer choices, we find that the equivalent expression is 2 12 55296 .
Examples
Understanding how to manipulate and simplify radical expressions is useful in various fields, such as engineering and physics, where calculations involving roots and exponents are common. For example, when calculating the impedance of an AC circuit, you might encounter expressions involving square roots and cube roots. Simplifying these expressions allows for easier calculations and a better understanding of the circuit's behavior. This skill is also valuable in computer graphics, where transformations and scaling often involve radical expressions.
