Chapter 3.1, Problem 18GP

Solution

The solutions of the equation ${\left(x+5\right)}^4=16$ .

  $x=-3$

Given:

The equation ${\left(x+5\right)}^4=16$ .

Concept Used:

• The even inter root of a number is defined for positive values only that are greater than or equal to 0.
• The odd integer root of all real numbers is defined.
• Power of a power property: ${\left(x^a\right)}^b=x^{a\cdot b}$
• The exponent form of the nth root of a number a is given by $\sqrt[n]{a}=a^{\frac{1}{n}}$

Calculation:

In order to solve the given equation ${\left(x+5\right)}^4=16$ , isolate the variable term on left side and simplify further using the exponent properties as shown below,

  $\begin{array}{c}{\left(x+5\right)}^4=16\\ {\left({\left(x+5\right)}^4\right)}^{\frac{1}{4}}={\left(16\right)}^{\frac{1}{4}}\\ x+5={\left(2^4\right)}^{\frac{1}{4}}\\ x+5=2\\ x+5-5=2-5\\ x=-3\end{array}$

Therefore, the solution of the equation is $x=-3$ .