To solve the equation [tex]\(12(1+w)^7 = 4\)[/tex] for [tex]\(w\)[/tex], given the constraint [tex]\(1+w>0\)[/tex], follow these steps:
1. Isolate the term involving the exponent:
[tex]\[
12(1+w)^7 = 4 \implies (1+w)^7 = \frac{4}{12}
\][/tex]
Simplify the fraction on the right-hand side:
[tex]\[
(1+w)^7 = \frac{1}{3}
\][/tex]
2. Take the seventh root of both sides to solve for [tex]\(1 + w\)[/tex]:
[tex]\[
1+w = \left(\frac{1}{3}\right)^{\frac{1}{7}}
\][/tex]
Let’s denote the seventh root of [tex]\(\frac{1}{3}\)[/tex] for simplicity:
[tex]\[
1+w = 0.854751
\][/tex]
3. Isolate [tex]\(w\)[/tex] by subtracting 1 from both sides:
[tex]\[
w = 0.854751 - 1
\][/tex]
[tex]\[
w = -0.145249
\][/tex]
4. Round the answer to the nearest thousandth:
[tex]\[
w \approx -0.145
\][/tex]
So, the value of [tex]\(w\)[/tex] rounded to the nearest thousandth is:
[tex]\[
w \approx -0.145
\][/tex]
