To determine which expression is equivalent to [tex]\(16^3\)[/tex], let's start by breaking it down step-by-step using properties of exponents.
1. Identify the base representation:
Observe that [tex]\(16\)[/tex] can be written as a power of [tex]\(2\)[/tex]. That is:
[tex]\[
16 = 2^4
\][/tex]
2. Express the given expression in terms of the base [tex]\(2\)[/tex]:
We need to find an equivalent expression for [tex]\(16^3\)[/tex]. Substitute [tex]\(16\)[/tex] with [tex]\(2^4\)[/tex]:
[tex]\[
16^3 = (2^4)^3
\][/tex]
3. Apply the exponentiation rules:
When raising a power to another power, you multiply the exponents. Specifically, [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Therefore:
[tex]\[
(2^4)^3 = 2^{4 \cdot 3}
\][/tex]
4. Simplify the exponent:
Calculate the product of the exponents:
[tex]\[
4 \cdot 3 = 12
\][/tex]
Thus, we have:
[tex]\[
2^{4 \cdot 3} = 2^{12}
\][/tex]
5. Identify the correct equivalent expression:
The expression equivalent to [tex]\(16^3\)[/tex] is [tex]\(2^{12}\)[/tex].
Hence, the correct answer is:
[tex]\[
2^{12}
\][/tex]
