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 simplifies to 6 2 . Thus, the correct answer is option 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 equivalent expression 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 simplify the expression, we can rewrite the radicals using exponents. Recall that n x = x n 1 . Therefore, we have 3 2 2 = 2 3 1 2 2 1 .
Applying the Quotient Rule Now, we can 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 by finding a common denominator for the fractions: 2 1 − 3 1 = 6 3 − 6 2 = 6 1 . So, the expression becomes 2 6 1 .
Converting Back to Radical Form Finally, we rewrite the expression using radicals again: 2 6 1 = 6 2 .
Final Answer Comparing our result with the given options, we see that 6 2 is one of the options. Therefore, the expression equivalent to 3 2 2 is 6 2 .
Examples
Understanding how to simplify expressions with radicals and exponents is useful in various fields, such as physics and engineering, where you often encounter complex formulas involving roots and powers. For example, when calculating the period of a pendulum, you might need to simplify an expression involving square roots. Similarly, in electrical engineering, when dealing with alternating current circuits, you may encounter expressions with fractional exponents. Knowing how to manipulate these expressions allows you to make accurate calculations and predictions.
