Which expression is equivalent to [tex][tex]$16^3$[/tex][/tex]?

A. [tex]$2^7$[/tex]
B. [tex]$2^{11}$[/tex]
C. [tex][tex]$2^{12}$[/tex][/tex]
D. [tex]$2^{64}$[/tex]



Answer :

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]