Chapter 3.1, Problem 58E

Solution

The solutions of the equation $x^6-34=181$ .

  $x\approx 2.45$

Given:

The equation $x^6-34=181$ .

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 $x^6-34=181$ , isolate the variable term on left side and simplify further using the exponent properties as shown below,

  $\begin{array}{c}{x}^6-34=181\\ x^6-34+34=181+34\\ x^6=215\\ {\left(x^6\right)}^{\frac{1}{6}}={\left(215\right)}^{\frac{1}{6}}\\ x={\left(215\right)}^{\frac{1}{6}}\\ x\approx 2.45\end{array}$

Therefore, the solution of the equation is $x\approx 2.45$ .