Select the equivalent expression.
$\sqrt[16]{\frac{m^8}{m^{-2}}}$
To simplify the expression \(\sqrt[16]{\frac{m^8}{m^{-2}}}\), we follow the rules of exponents. First, focus on the expression inside the radical. Using the quotient rule for exponents, which states \(\frac{a^x}{a^y} = a^{x-y}\), we can simplify the fraction:
\[\frac{m^8}{m^{-2}} = m^{8 - (-2)} = m^{8+2} = m^{10}\]
Now that the expression is simplified to \(m^{10}\), we substitute it back into the original expression to get \(\sqrt[16]{m^{10}}\). A radical of the form \(\sqrt[n]{x}\) can be written as \(x^{1/n}\). Applying this rule:
\[\sqrt[16]{m^{10}} = (m^{10})^{\frac{1}{16}}\]
Finally, we use the power of a power rule for exponents, which states \((a^x)^y = a^{x \cdot y}\):
\[m^{10 \cdot \frac{1}{16}} = m^{\frac{10}{16}}\]
Simplifying the fraction \(\frac{10}{16}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2, we obtain:
\[\frac{10}{16} = \frac{5}{8}\]
Thus, the simplified expression is \(m^{\frac{5}{8}}\). This corresponds to the third choice.
Understanding exponent rules is essential in fields like computer science and engineering. For instance, when analyzing the complexity of algorithms, you might encounter expressions like \(n^2/n^{1/2}\), which simplifies to \(n^{3/2}\). Similarly, when dealing with signal processing, exponents are used to express energy attenuation or amplification. Mastering these simplifications helps in performing quick calculations and understanding the growth patterns of various systems.