To express [tex]\( 8^4 \)[/tex] as a power of 2, follow these steps:
1. Understand the base in terms of powers of 2:
The number 8 can be written as a power of 2. Specifically,
[tex]\[
8 = 2^3
\][/tex]
2. Express the exponentiation [tex]\( 8^4 \)[/tex] using the base representation in terms of a power of 2:
Substitute [tex]\( 8 \)[/tex] with [tex]\( 2^3 \)[/tex] in [tex]\( 8^4 \)[/tex]:
[tex]\[
8^4 = (2^3)^4
\][/tex]
3. Simplify the expression:
When raising a power to another power, you multiply the exponents. So,
[tex]\[
(2^3)^4 = 2^{3 \cdot 4}
\][/tex]
4. Calculate the new exponent:
Multiply the exponents together:
[tex]\[
3 \cdot 4 = 12
\][/tex]
5. Write the final result:
Therefore,
[tex]\[
8^4 = 2^{12}
\][/tex]
So, [tex]\( 8^4 \)[/tex] expressed as a power of 2 is [tex]\( 2^{12} \)[/tex].
Additionally, if you compute [tex]\( 2^{12} \)[/tex], you get:
[tex]\[
2^{12} = 4096
\][/tex]
Thus, the detailed breakdown confirms that
[tex]\[
8^4 = 2^{12} = 4096
\][/tex]
