Which expression is equivalent to $\frac{\sqrt{2}}{\sqrt[3]{2}}$ ?
A. $\frac{1}{4}$
B. $\sqrt[6]{2}$
C. $\sqrt{2}$
D. $\frac{\sqrt{2}}{2}$
Answer (2)
The expression 3 2 2 can be rewritten as 6 2 using exponents and the quotient rule for exponents. The correct answer among the options given is B . 6 2 .
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Rewrite the expression using exponents: 3 2 2 = 2 3 1 2 2 1 .
Apply the quotient rule for exponents: 2 3 1 2 2 1 = 2 2 1 − 3 1 .
Simplify the exponent: 2 1 − 3 1 = 6 1 .
Rewrite the expression using radicals: 2 6 1 = 6 2 . The final answer is 6 2 .
Explanation
Understanding the Problem We are given the expression 3 2 2 and asked to find an equivalent expression from the options: 4 1 , 6 2 , 2 , 2 2 .
Rewriting with Exponents To solve this, we will rewrite the expression using exponents. Recall that n x = x n 1 . Therefore, 2 = 2 2 1 and 3 2 = 2 3 1 . So, the given expression can be written as 2 3 1 2 2 1 .
Applying the Quotient Rule Now, we use the quotient rule for exponents, which states that a n a m = a m − n . Applying this rule, we get 2 3 1 2 2 1 = 2 2 1 − 3 1 .
Simplifying the Exponent Next, we simplify the exponent. We have 2 1 − 3 1 = 6 3 − 6 2 = 6 1 . Therefore, the expression simplifies to 2 6 1 .
Converting Back to Radical Form Finally, we rewrite the expression using radicals. Since x n 1 = n x , we have 2 6 1 = 6 2 .
Final Answer Comparing our result with the given options, we see that the equivalent expression is 6 2 .
Examples
Understanding fractional exponents and radicals is crucial in various fields, such as physics and engineering, where you might encounter expressions involving roots and powers when dealing with wave functions or signal processing. For instance, when analyzing the behavior of sound waves or electromagnetic waves, you often need to simplify expressions containing radicals and exponents to understand the wave's properties, such as its amplitude or frequency. Simplifying these expressions allows engineers and physicists to make accurate predictions and design efficient systems.
