To find which expression is equivalent to [tex]\( 16^3 \)[/tex], we can break down the problem step-by-step.
Firstly, let's recall that [tex]\( 16 \)[/tex] can be expressed as a power of [tex]\( 2 \)[/tex]:
[tex]\[ 16 = 2^4 \][/tex]
Now, raise [tex]\( 16 \)[/tex] to the power of [tex]\( 3 \)[/tex]:
[tex]\[ (2^4)^3 \][/tex]
Using the properties of exponents, we know that [tex]\((a^m)^n = a^{mn}\)[/tex]:
[tex]\[ (2^4)^3 = 2^{4 \cdot 3} \][/tex]
[tex]\[ 2^{4 \cdot 3} = 2^{12} \][/tex]
Therefore, [tex]\( 16^3 \)[/tex] is equivalent to [tex]\( 2^{12} \)[/tex].
Among the given options:
[tex]\[ 2^7 \][/tex]
[tex]\[ 2^{11} \][/tex]
[tex]\[ 2^{12} \][/tex]
[tex]\[ 2^{64} \][/tex]
The correct equivalent expression for [tex]\( 16^3 \)[/tex] is [tex]\( 2^{12} \)[/tex].
