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 equivalent expression to 3 2 4 6 is 2 12 55296 , which corresponds to option C. To arrive at this result, we rewrote the expression using fractional exponents, expressed both the numerator and denominator with a common exponent, and then simplified the result. Finally, we rationalized the denominator to confirm the equivalent expression.
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Rewrite the given expression using fractional exponents: 3 2 4 6 = 2 1/3 6 1/4 .
Express the numerator and denominator with a common exponent: 2 4/12 6 3/12 = 12 2 4 12 6 3 = 12 2 4 6 3 .
Simplify the fraction inside the radical: 12 2 4 6 3 = 12 16 216 = 12 2 27 .
Rationalize the denominator: 12 2 12 27 = 2 12 55296 .
The equivalent expression 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.
Fractional Exponents First, let's rewrite the given expression using fractional exponents: 3 2 4 6 = 2 1/3 6 1/4
Common Exponent To compare the expression with the given options, we need to find a common exponent. The least common multiple of 4 and 3 is 12. So, we rewrite the expression with the exponent 1/12: 2 1/3 6 1/4 = 2 4/12 6 3/12 = 12 2 4 12 6 3 = 12 2 4 6 3
Simplifying the Fraction Now, let's simplify the fraction inside the radical: 12 2 4 6 3 = 12 16 216 = 12 8 108 = 12 4 54 = 12 2 27
Rationalizing the Denominator We can rewrite the expression as: 12 2 27 = 12 2 12 27 To rationalize the denominator, we multiply both the numerator and denominator by 12 2 11 : 12 2 12 27 ⋅ 12 2 11 12 2 11 = 12 2 12 12 27 ⋅ 2 11 = 2 12 27 ⋅ 2 11 Since 2 11 = 2048 , we have: 2 12 27 ⋅ 2048 = 2 12 55296
Finding the Equivalent Expression Comparing the simplified expression with the given options, we find that the equivalent expression is 2 12 55296 .
Final Answer Therefore, the expression equivalent to 3 2 4 6 is 2 12 55296 .
Examples
Understanding how to simplify and manipulate radical expressions is useful in various fields, such as engineering and physics, where complex calculations involving roots and exponents are common. For example, when calculating the impedance of an electrical circuit, you might encounter expressions involving radicals that need to be simplified to find the overall impedance. Similarly, in physics, when dealing with wave equations or quantum mechanics, simplifying radical expressions can make calculations more manageable and lead to a clearer understanding of the underlying phenomena. This skill is also crucial in computer graphics for calculating distances and transformations.
