Question

1. $x^5 = -112$

Explanation:

Step1: Take the fifth root of both sides

To solve for \( x \) in the equation \( x^{5}=-112 \), we take the fifth root of both sides. The formula for the \( n \)-th root of a number \( a \) is \( \sqrt[n]{a} \), so for \( n = 5 \) and \( a=-112 \), we have \( x=\sqrt[5]{-112} \). We can also write this as \( x = (-112)^{\frac{1}{5}} \).

Step2: Calculate the value

Using a calculator to find the fifth root of \(- 112\), we know that \( \sqrt[5]{112}\approx2.47 \) (since \( 2.47^{5}\approx112 \)), so \( \sqrt[5]{-112}\approx - 2.47 \) (because the fifth root of a negative number is negative).

Answer:

\( x\approx - 2.47 \) (or \( x = \sqrt[5]{-112}\))