Certainly! Let's simplify each expression step-by-step.
### Expression a: [tex]\(2^3 \cdot 3^2\)[/tex]
[tex]\(2^3 = 8\)[/tex] and [tex]\(3^2 = 9\)[/tex]
So, [tex]\(2^3 \cdot 3^2 = 8 \cdot 9 = 72\)[/tex].
Thus, the simplified result is [tex]\(72\)[/tex].
### Expression b: [tex]\(2^2 + 2^3\)[/tex]
[tex]\(2^2 = 4\)[/tex] and [tex]\(2^3 = 8\)[/tex]
So, [tex]\(2^2 + 2^3 = 4 + 8 = 12\)[/tex].
Thus, the simplified result is [tex]\(12\)[/tex].
### Expression c: [tex]\(2^2 \cdot 2^3\)[/tex]
Using the properties of exponents: [tex]\(a^m \cdot a^n = a^{m+n}\)[/tex],
[tex]\(2^2 \cdot 2^3 = 2^{2+3} = 2^5\)[/tex].
So, [tex]\(2^5 = 32\)[/tex].
Thus, the simplified result is [tex]\(32\)[/tex].
### Expression d: [tex]\(\left(2^2\right)^3\)[/tex]
Using the properties of exponents: [tex]\((a^m)^n = a^{mn}\)[/tex],
[tex]\(\left(2^2\right)^3 = 2^{2 \cdot 3} = 2^6\)[/tex].
So, [tex]\(2^6 = 64\)[/tex].
Thus, the simplified result is [tex]\(64\)[/tex].
### Expression e: [tex]\(\frac{2^2}{2^3}\)[/tex]
Using the properties of exponents: [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex],
[tex]\(\frac{2^2}{2^3} = 2^{2-3} = 2^{-1}\)[/tex].
So, [tex]\(2^{-1} = \frac{1}{2}\)[/tex].
Thus, the simplified result is [tex]\(0.5\)[/tex].
### Expression f: [tex]\(2^2 - 2^4\)[/tex]
[tex]\(2^2 = 4\)[/tex] and [tex]\(2^4 = 16\)[/tex]
So, [tex]\(2^2 - 2^4 = 4 - 16 = -12\)[/tex].
Thus, the simplified result is [tex]\(-12\)[/tex].
In summary, the simplified results for the expressions are:
a. [tex]\(72\)[/tex]
b. [tex]\(12\)[/tex]
c. [tex]\(32\)[/tex]
d. [tex]\(64\)[/tex]
e. [tex]\(0.5\)[/tex]
f. [tex]\(-12\)[/tex]
