To solve the equation [tex]\(2^{x+3} + 2^{x+1} = 80\)[/tex], let's break down the process step by step.
1. Simplify the Exponents:
Notice that we can rewrite the terms to make the exponents more manageable. We begin by factoring out a common term: [tex]\[
2^{x+3} = 2^x \cdot 2^3 = 2^x \cdot 8
\][/tex] and [tex]\[
2^{x+1} = 2^x \cdot 2 = 2^x \cdot 2
\][/tex]
So the equation now looks like: [tex]\[
8 \cdot 2^x + 2 \cdot 2^x = 80
\][/tex]
2. Factor Out [tex]\(2^x\)[/tex]:
Combine like terms by factoring out [tex]\(2^x\)[/tex]: [tex]\[
2^x (8 + 2) = 80
\][/tex] Simplify the expression inside the parentheses: [tex]\[
2^x \cdot 10 = 80
\][/tex]
3. Solve for [tex]\(2^x\)[/tex]:
To isolate [tex]\(2^x\)[/tex], divide both sides of the equation by 10: [tex]\[
2^x = \frac{80}{10}
\][/tex] Simplify the right side: [tex]\[
2^x = 8
\][/tex]
4. Express 8 as a Power of 2:
Recognize that 8 can be written as a power of 2: [tex]\[
8 = 2^3
\][/tex] Therefore: [tex]\[
2^x = 2^3
\][/tex]
5. Equate the Exponents:
Since the bases are the same, we can set the exponents equal to each other: [tex]\[
x = 3
\][/tex]
So, the solution to the equation [tex]\(2^{x+3} + 2^{x+1} = 80\)[/tex] is [tex]\(x = 3\)[/tex].
