To solve the equation:
[tex]\[
\left(x^2 - 9\right)^{\frac{1}{4}} + 7 = 5
\][/tex]
we need to isolate [tex]\(x\)[/tex]. Let's go through the steps:
1. Subtract 7 from both sides to isolate the fourth root term:
[tex]\[
\left(x^2 - 9\right)^{\frac{1}{4}} = 5 - 7
\][/tex]
which simplifies to:
[tex]\[
\left(x^2 - 9\right)^{\frac{1}{4}} = -2
\][/tex]
2. Raise both sides to the fourth power to eliminate the rational exponent:
[tex]\[
\left(\left(x^2 - 9\right)^{\frac{1}{4}}\right)^4 = (-2)^4
\][/tex]
This gives us:
[tex]\[
x^2 - 9 = 16
\][/tex]
3. Solve for [tex]\(x^2\)[/tex]:
[tex]\[
x^2 - 9 = 16
\][/tex]
Add 9 to both sides:
[tex]\[
x^2 = 25
\][/tex]
4. Solve for [tex]\(x\)[/tex] by taking the square root of both sides:
[tex]\[
x = \pm 5
\][/tex]
Given that [tex]\(\left(x^2 - 9\right)^{\frac{1}{4}} ≥ 0\)[/tex], notice that
[tex]\[
\left(x^2 - 9\right)^{\frac{1}{4}} = -2
\][/tex]
is not possible, as a fourth root cannot be negative. Therefore, there are no values of [tex]\(x\)[/tex] that satisfy the original equation.
Thus, the equation has
[tex]\[
\boxed{ \ \ }
\][/tex]
(no solutions).
