Chapter 7.7, Problem 68HP

To determine

To Find:

If Lola is correct and explain.

Answer to Problem 68HP

Yes, Lola is correct.

Explanation of Solution

Given Information:

Lola is working to solve ${\left(x+5\right)}^{\frac{1}{4}}=-4$ . She said that she could tell there was no real solution without even working the problem.

Concept Used:

• Definition of nth root:
• Words : For any real numbers a and b , and any positive integer n , if $a^n=b$ then a is the nth root of b . That is $a=\sqrt[n]{b}$
• $\begin{array}{l}\sqrt[n]{b}\\ Here\\ b=radicand\\ n=index\\ \sqrt{}=radicalsign\end{array}$
• The value of radical is always non-negative , when it has real roots.

Calculation:

  $\begin{array}{l}\text{Consider the equation:}\\ {\left(x+5\right)}^{\frac{1}{4}}=-4\\ \mathrm{Sin}ce\\ {\left(x+5\right)}^{\frac{1}{4}}\ge 0\\ \text{As we know that the value of radical is always non-negative when it has real roots}\text{.}\\ \text{But here , the value of the radical is -4, which is negative , so this equation }\\ \text{does not have any real solution}\text{.}\\ \text{Or we can that the equation has no solution}\text{.}\\ \text{Hence, Lola is correct}\text{.}\end{array}$