Which expression is equivalent to $144^{\frac{3}{2}}$?
The given problem asks for the equivalent expression of \(144^{\frac{3}{2}}\). An expression with a rational exponent \(a^{\frac{m}{n}}\) can be expressed using roots and powers according to the rule:
\[a^{\frac{m}{n}} = (\sqrt[n]{a})^m\]
Here, \(a = 144\), \(m = 3\), and \(n = 2\). Thus, the expression becomes:
\[144^{\frac{3}{2}} = (\sqrt{144})^3\]
Now, calculate the root part first. Since \(144\) is a perfect square, we find its square root:
\[\sqrt{144} = 12\]
Next, substitute this value into the expression and raise it to the third power:
\[(\sqrt{144})^3 = 12^3\]
Now, compute \(12^3\):
\[12^3 = 12 \times 12 \times 12 = 144 \times 12 = 1728\]
Comparing our result, \(1728\), with the given choices:
◯ 216
◯ 1,728
◯ \(\sqrt[3]{12}\)
◯ \(\sqrt[3]{72}\)
The value \(1728\) matches the second choice. Therefore, the equivalent expression is \(\boxed{1728}\).
Understanding how to handle rational exponents is essential for fields like physics and engineering, where you often encounter formulas involving powers of variables. For example, in calculating the period of a planet's orbit using Kepler's Third Law, you might need to compute expressions where a value is raised to a fractional power, such as \(\frac{3}{2}\). Mastering these conversions allows you to simplify complex physical models and perform precise calculations in your daily scientific or mathematical endeavors.