Question

solve the equation. round answers to four decimal places.
$(1 + r)^5 = 3.8$
$r \approx \square$
(round to four decimal places as needed.)

Explanation:

Step1: Take the 5th root of both sides

To solve for \(1 + r\), we take the 5th root of both sides of the equation \((1 + r)^5 = 3.8\). Using the property of exponents, \((a^n)^{\frac{1}{n}}=a\) for \(a\geq0\) and \(n\) a positive integer, we get \(1 + r = 3.8^{\frac{1}{5}}\) or \(1 + r=\sqrt[5]{3.8}\).

Step2: Calculate the 5th root of 3.8

We can calculate \(3.8^{\frac{1}{5}}\) using a calculator. \(3.8^{\frac{1}{5}}\approx1.3095\) (using a calculator to compute the 5th root of 3.8).

Step3: Solve for r

Subtract 1 from both sides of the equation \(1 + r\approx1.3095\) to get \(r\approx1.3095 - 1\).

Step4: Compute the value of r

\(r\approx0.3095\) (after subtracting 1 from 1.3095).

Answer:

\(r\approx\boxed{0.3095}\)