Simplify $\frac{\sqrt{2}}{\sqrt[3]{2}}$.
A. $2^{\frac{1}{6}}$
B. $2^{\frac{1}{3}}$
C. $2^{\frac{5}{6}}$
D. $2^{\frac{3}{2}}$
Answer (1)
Rewrite the radicals as exponents: 2 = 2 2 1 and 3 2 = 2 3 1 .
Apply the quotient rule of exponents: 2 3 1 2 2 1 = 2 2 1 − 3 1 .
Subtract the exponents: 2 1 − 3 1 = 6 1 .
The simplified expression is 2 6 1 .
Explanation
Understanding the Problem We are asked to simplify the expression 3 2 2 . To do this, we will rewrite the radicals as exponents and then use the properties of exponents to simplify.
Rewriting Radicals as Exponents First, we rewrite the radicals as exponents. Recall that n x = x n 1 . Therefore, 2 = 2 2 1 and 3 2 = 2 3 1 .
Applying the Quotient Rule of Exponents Now we can rewrite the expression as 2 3 1 2 2 1 . To simplify this expression, we use the property a n a m = a m − n . Therefore, we have 2 2 1 − 3 1 .
Subtracting the Exponents Now we need to find a common denominator to subtract the fractions in the exponent. The common denominator of 2 and 3 is 6. So we have 2 1 − 3 1 = 6 3 − 6 2 = 6 1 .
Final Answer Therefore, the simplified expression is 2 6 1 .
Examples
Understanding exponents and radicals is crucial in many scientific fields. For example, in physics, the energy of a photon is related to its frequency by the equation E = h f , where h is Planck's constant. If we express the frequency as a function of wavelength, f = λ c , where c is the speed of light and λ is the wavelength, and then consider the square root of the energy, we might encounter expressions involving fractional exponents similar to the one in this problem. Simplifying such expressions allows for easier calculations and a better understanding of the relationships between different physical quantities.
