Which expression is equivalent to $\frac{\sqrt{2}}{\sqrt[3]{2}}$ ?
To determine the expression equivalent to \(\frac{\sqrt{2}}{\sqrt[3]{2}}\), we first express the square root and the cube root as exponents of 2. Recall that the square root of a number \(x\) is \(x^{1/2}\) and the cube root of \(x\) is \(x^{1/3}\). Therefore, the numerator is \(\sqrt{2} = 2^{1/2}\) and the denominator is \(\sqrt[3]{2} = 2^{1/3}\).
Now, we rewrite the given expression \(\frac{\sqrt{2}}{\sqrt[3]{2}}\) using these power forms:
\[\frac{\sqrt{2}}{\sqrt[3]{2}} = \frac{2^{1/2}}{2^{1/3}}.\]
According to the quotient rule for exponents, \(\frac{x^m}{x^n} = x^{m-n}\). Applying this here gives us:
\[\frac{2^{1/2}}{2^{1/3}} = 2^{1/2 - 1/3}.\]
Next, we subtract the exponents:
\[\frac{1}{2} - \frac{1}{3} = \frac{3}{6} - \frac{2}{6} = \frac{1}{6}.\]
So, the expression simplifies to \(2^{1/6}\). Converting this back to radical notation, \(2^{1/6}\) is equivalent to the sixth root of 2, written as \(\sqrt[6]{2}\). This corresponds to the second option provided.
In engineering and science, radicals are often used to express dimensions, growth rates, or signal intensities. For example, when calculating the geometric mean of growth factors or scaling objects in different dimensions, one might encounter products or quotients of various radicals. Converting these to exponents allows for much easier manipulation and simplification, just as we did here by converting roots to fractions. This method is a cornerstone for simplifying complex expressions in physics and calculus.