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Solve the equation [tex]$6(1+x)^5=2$[/tex]. Assume that [tex]$1+x\ \textgreater \ 0$[/tex]. Round your answer to the nearest thousandth (3 decimal places).

[tex]x =[/tex]


Sagot :

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]