To determine which expression is equivalent to [tex]\(16^3\)[/tex], we first need to calculate [tex]\(16^3\)[/tex].
1. Calculate [tex]\(16^3\)[/tex]:
[tex]\[
16^3 = 16 \times 16 \times 16 = 4096
\][/tex]
Next, we need to evaluate each of the given expressions and see which one equals [tex]\(4096\)[/tex].
2. Evaluate [tex]\(2^7\)[/tex]:
[tex]\[
2^7 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 128
\][/tex]
[tex]\(2^7 = 128\)[/tex], which is not equal to [tex]\(4096\)[/tex].
3. Evaluate [tex]\(2^{11}\)[/tex]:
[tex]\[
2^{11} = 2 \times 2 \times \ldots \times 2 \quad (\text{11 times})
\][/tex]
Calculating step-by-step:
[tex]\[
2^2 = 4, \quad
2^3 = 8, \quad
2^4 = 16, \quad
2^5 = 32
\][/tex]
[tex]\[
2^6 = 64, \quad
2^7 = 128, \quad
2^8 = 256, \quad
2^9 = 512
\][/tex]
[tex]\[
2^{10} = 1024, \quad
2^{11} = 2048
\][/tex]
[tex]\(2^{11} = 2048\)[/tex], which is not equal to [tex]\(4096\)[/tex].
4. Evaluate [tex]\(2^{12}\)[/tex]:
[tex]\[
2^{12} = 2 \times 2 \times \ldots \times 2 \quad (\text{12 times})
\][/tex]
Calculating step-by-step:
[tex]\[
2^2 = 4, \quad
2^3 = 8, \quad
2^4 = 16, \quad
2^5 = 32
\][/tex]
[tex]\[
2^6 = 64, \quad
2^7 = 128, \quad
2^8 = 256, \quad
2^9 = 512
\][/tex]
[tex]\[
2^{10} = 1024, \quad
2^{11} = 2048, \quad
2^{12} = 4096
\][/tex]
[tex]\(2^{12} = 4096\)[/tex], which is indeed equal to [tex]\(4096\)[/tex].
5. Evaluate [tex]\(2^{64}\)[/tex]:
This number is extremely large:
[tex]\[
2^{64} = 18446744073709551616
\][/tex]
[tex]\(2^{64}\)[/tex] is much larger than [tex]\(4096\)[/tex].
So, the expression that is equivalent to [tex]\(16^3\)[/tex] is:
[tex]\[
2^{12}
\][/tex]
