To determine the common base that can be used to rewrite each side of the equation [tex]\(2^{x+3} - 3 = 5\)[/tex], follow these steps:
1. Isolate the exponential term:
Start by isolating [tex]\(2^{x+3}\)[/tex]. Add 3 to both sides of the equation:
[tex]\[
2^{x+3} - 3 + 3 = 5 + 3
\][/tex]
This simplifies to:
[tex]\[
2^{x+3} = 8
\][/tex]
2. Express 8 as a power of 2:
Notice that 8 can be written as [tex]\(2^3\)[/tex]. This is because:
[tex]\[
8 = 2^3
\][/tex]
3. Rewrite the equation with a common base:
Now that we know [tex]\(8\)[/tex] is [tex]\(2^3\)[/tex], we can rewrite the equation [tex]\(2^{x+3} = 8\)[/tex] using the common base [tex]\(2\)[/tex]:
[tex]\[
2^{x+3} = 2^3
\][/tex]
4. Simplify the exponents:
Since the bases are the same, we can set the exponents equal to each other:
[tex]\[
x + 3 = 3
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
Subtract 3 from both sides to isolate [tex]\(x\)[/tex]:
[tex]\[
x = 3 - 3
\][/tex]
[tex]\[
x = 0
\][/tex]
The common base that we used to rewrite each side of the equation is:
[tex]\[
\boxed{2}
\][/tex]
This ensures the equation holds true when solved in terms of exponential form.
