Sure, let's break down the problem step by step.
We are given the expression:
[tex]\[
\frac{2^{a+3} - 2^{a+1}}{3 \times 2^a}
\][/tex]
First, let's look at the numerator [tex]\(2^{a+3} - 2^{a+1}\)[/tex] and simplify it:
1. Rewrite the terms using the properties of exponents:
[tex]\[
2^{a+3} = 2^a \cdot 2^3 = 2^a \cdot 8
\][/tex]
[tex]\[
2^{a+1} = 2^a \cdot 2 = 2^a \cdot 2
\][/tex]
2. Substitute these back into the numerator:
[tex]\[
2^{a+3} - 2^{a+1} = 2^a \cdot 8 - 2^a \cdot 2
\][/tex]
3. Factor out [tex]\(2^a\)[/tex] from both terms:
[tex]\[
2^a \cdot 8 - 2^a \cdot 2 = 2^a (8 - 2) = 2^a \cdot 6
\][/tex]
So the numerator simplifies to:
[tex]\[
2^a \cdot 6
\][/tex]
Now let's rewrite the expression with the simplified numerator:
[tex]\[
\frac{2^a \cdot 6}{3 \times 2^a}
\][/tex]
Next, let's simplify the entire fraction:
1. Notice that [tex]\(2^a\)[/tex] in the numerator and the denominator can cancel out:
[tex]\[
\frac{6 \cdot 2^a}{3 \cdot 2^a} = \frac{6}{3}
\][/tex]
2. Simplify [tex]\( \frac{6}{3} \)[/tex]:
[tex]\[
\frac{6}{3} = 2
\][/tex]
Therefore, the simplified form of the expression is:
[tex]\[
2
\][/tex]
So, the detailed, step-by-step simplification process for the given expression [tex]\(\frac{2^{a+3} - 2^{a+1}}{3 \times 2^a}\)[/tex] yields:
[tex]\[
2
\][/tex]
