Which of the following is equivalent to $16^{\frac{3}{4}}$?
The problem asks to evaluate the expression \(16^{\frac{3}{4}}\) and identify which of the provided options—6, 8, 12, or 64—is equivalent to it. To solve this, we can use the properties of exponents. Specifically, we know that if we can express the base as a power of a smaller number, we can simplify the expression significantly. The expression is \(16^{\frac{3}{4}}\).
We observe that 16 can be written as a power of 2: \[16 = 2 \times 2 \times 2 \times 2 = 2^4.\] Substituting this into our expression, we get: \[16^{\frac{3}{4}} = (2^4)^{\frac{3}{4}}.\] Using the power of a power property, which states that \((a^n)^m = a^{n \times m}\), we can simplify the exponent: \[2^{4 \times \frac{3}{4}} = 2^3.\]
Now that we have the expression simplified to \(2^3\), we can easily calculate its value: \[2^3 = 2 \times 2 \times 2 = 8.\] Comparing this result to our options (6, 8, 12, 64), we see that the correct value is 8.
Understanding exponents is crucial in various real-life scenarios, such as calculating compound interest. If an investment grows by a certain rate over several periods, the formula involves exponentiation. For example, if an initial amount grows by a certain factor after a fractional period, you use fractional exponents to determine the value, similar to how we evaluated 16 raised to a fractional power here to find the precise growth result.