Which of the following is equivalent to $60^{\frac{1}{2}}$ ?
To determine which expression is equivalent to \(60^{\frac{1}{2}}\), we must look at the mathematical rule for fractional exponents. The rule states that for any non-negative base \(x\) and a positive integer \(n\), the expression \(x^{\frac{1}{n}}\) is equivalent to the \(n\)-th root of \(x\), which is written as \(\sqrt[n]{x}\).
In this specific problem, we have the expression \(60^{\frac{1}{2}}\). Here, the base \(x\) is \(60\) and the exponent is \(\frac{1}{2}\). By applying the fractional exponent rule where \(n=2\), we get:
\[60^{\frac{1}{2}} = \sqrt[2]{60} = \sqrt{60}.\]
This square root expression is the standard form of a power with a fractional exponent of \(\frac{1}{2}\).
Now, we compare our result with the given choices:
The result \(\sqrt{60}\) matches the second option. Therefore, \(60^{\frac{1}{2}}\) is equivalent to \(\sqrt{60}\).
Fractional exponents are used extensively in fields like physics and engineering to represent growth rates and scaling factors. For example, if an engineer is calculating the stress on a material or the signal strength in a telecommunications system, they might encounter square root relationships expressed as \(x^{0.5}\). Understanding that this is equivalent to \(\sqrt{x}\) helps them simplify complex formulas into more manageable, intuitive terms, making it easier to predict how systems will perform under different conditions.