To solve the equation [tex]\(6(1+x)^5 = 2\)[/tex], we will follow these steps:
1. Isolate the term involving [tex]\(x\)[/tex]:
Start by dividing both sides of the equation by 6 to isolate [tex]\((1 + x)^5\)[/tex]:
[tex]\[
6(1 + x)^5 = 2 \implies (1 + x)^5 = \frac{2}{6} = \frac{1}{3}
\][/tex]
2. Take the fifth root of both sides:
We want to solve for [tex]\(1 + x\)[/tex], so we'll take the fifth root of both sides:
[tex]\[
(1 + x) = \left(\frac{1}{3}\right)^{\frac{1}{5}}
\][/tex]
This simplifies to:
[tex]\[
1 + x = \left(\frac{1}{3}\right)^{0.2}
\][/tex]
3. Approximate the value:
Calculate the approximate value of [tex]\(\left(\frac{1}{3}\right)^{0.2}\)[/tex]. This can typically be done with a calculator:
[tex]\[
\left(\frac{1}{3}\right)^{0.2} \approx 0.806
\][/tex]
So we have:
[tex]\[
1 + x \approx 0.806
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
Subtract 1 from both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x \approx 0.806 - 1 = -0.194
\][/tex]
Therefore, the solution to the equation [tex]\(6(1+x)^5 = 2\)[/tex] rounded to the nearest thousandth is:
[tex]\[
x \approx -0.194
\][/tex]
