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)
Rewrite the expression using fractional exponents: 3 2 4 6 = 2 1/3 6 1/4 .
Rewrite with a common denominator: 2 1/3 6 1/4 = 2 4/12 6 3/12 .
Rewrite using radicals with index 12: 2 4/12 6 3/12 = 12 2 4 12 6 3 = 12 16 12 216 .
Simplify and match the answer choices: 12 16 12 216 = 12 2 27 = 2 12 55296 . The final answer is 2 12 55296 .
Explanation
Problem Analysis We are given the expression 3 2 4 6 and asked to find an equivalent expression from the given options. Our approach involves converting the radicals to fractional exponents, finding a common denominator for the exponents, and simplifying the expression to match one of the answer choices.
Fractional Exponents First, we rewrite the given expression using fractional exponents: 3 2 4 6 = 2 1/3 6 1/4
Common Denominator Next, we find a common denominator for the exponents. The least common multiple of 3 and 4 is 12. So, we rewrite the exponents with a denominator of 12: 2 1/3 6 1/4 = 2 4/12 6 3/12
Radicals with Index 12 Now, we rewrite the expression using radicals with index 12: 2 4/12 6 3/12 = 12 2 4 12 6 3 = 12 16 12 216
Combining Radicals We combine the radicals into a single radical: 12 16 12 216 = 12 16 216 = 12 8 108 = 12 4 54 = 12 2 27
Matching Answer Choices To match the form of the answer choices, which have a denominator of 2 or 3 outside the radical, we want to rewrite the expression with a denominator of 2. Thus, we want to express 12 2 27 as 2 12 x for some x. This means 12 2 27 = 12 2 12 12 x . Therefore, 2 27 = 2 12 x , so x = 27 ⋅ 2 11 = 27 ⋅ 2048 = 55296 . Thus, the expression is 2 12 55296 .
Final Answer Comparing the simplified expression with the given options, we find that the equivalent expression is 2 12 55296 .
Examples
Understanding how to simplify and manipulate 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 electrical circuit or determining the energy levels of an electron in a quantum mechanical system, simplifying radical expressions can make the calculations more manageable and lead to more accurate results. This skill also helps in understanding the relationships between different physical quantities and making predictions about their behavior.
The expression equivalent to 3 2 4 6 is 2 12 55296 . After rewriting and simplifying the radicals and fractional exponents, we identified the matching option from the choices. Therefore, the answer is oxed{\frac{\sqrt[12]{55296}}{2}} .
;
