Question

solve.
\\(sqrt7{7x - 6} = sqrt7{9x + 14}\\)
select the correct choice below and fill in any answer boxes in your choice.
a. the solution set is {}.
(simplify your answer. use a comma to separate answers as needed.)
b. the solution is not a real number.

Explanation:

Step1: Raise both sides to the 7th power

Since the nth root function (for odd n, like 7 here) is one - to - one, if \(\sqrt[7]{a}=\sqrt[7]{b}\), then \(a = b\). So we raise both sides of the equation \(\sqrt[7]{7x - 6}=\sqrt[7]{9x + 14}\) to the 7th power.
\((\sqrt[7]{7x - 6})^7=(\sqrt[7]{9x + 14})^7\)
Which simplifies to \(7x-6 = 9x + 14\)

Step2: Solve for x

Subtract \(7x\) from both sides:
\(7x-6-7x=9x + 14-7x\)
\(-6 = 2x+14\)
Subtract 14 from both sides:
\(-6-14=2x+14 - 14\)
\(-20 = 2x\)
Divide both sides by 2:
\(x=\frac{-20}{2}=- 10\)

Step3: Check the solution

We need to check if the solution makes the expressions inside the 7th roots defined (since the domain of \(\sqrt[7]{u}\) is all real numbers \(u\), because the 7th root of a negative number is defined in the real number system).
For \(x=-10\), the left - hand side: \(7x-6=7\times(-10)-6=-70 - 6=-76\)
The right - hand side: \(9x + 14=9\times(-10)+14=-90 + 14=-76\)
Both expressions inside the 7th roots are real numbers, so \(x = - 10\) is a valid solution.

Answer:

A. The solution set is \(\{-10\}\)