To solve the given problem, we need to find the value of [tex]\( 16^{\frac{3}{4}} \)[/tex] and determine which of the given options is closest to this value.
Let's break it down step by step:
1. Identify the Base and Exponent:
The base is 16 and the exponent is [tex]\( \frac{3}{4} \)[/tex].
2. Rewrite the Base:
We know that 16 is a power of 2, specifically:
[tex]\[
16 = 2^4
\][/tex]
3. Apply the Exponent:
We need to apply the exponent [tex]\( \frac{3}{4} \)[/tex] to 16:
[tex]\[
16^{\frac{3}{4}} = (2^4)^{\frac{3}{4}}
\][/tex]
4. Use Power of a Power Property:
The property of exponents [tex]\( (a^m)^n = a^{mn} \)[/tex] tells us that we can multiply the exponents:
[tex]\[
(2^4)^{\frac{3}{4}} = 2^{4 \cdot \frac{3}{4}} = 2^3
\][/tex]
5. Calculate the Final Value:
We calculate [tex]\( 2^3 \)[/tex]:
[tex]\[
2^3 = 8
\][/tex]
Thus, [tex]\( 16^{\frac{3}{4}} \)[/tex] evaluates to 8.
6. Compare with the Given Options:
The possible answers provided are 6, 8, 12, and 64. Since our calculated value is 8, the value that is equivalent to [tex]\( 16^{\frac{3}{4}} \)[/tex] is:
[tex]\[
\boxed{8}
\][/tex]
