Question

solve the equation with rational exponents.$left(x^{2}-x-29
ight)^{3/4}-15=12$select the correct choice below and, if necessary, fill in the answer boxa. the solution(s) is(are) $x =$(simplify your answer. use a comma to separate answers as neb. there is no solution.

Explanation:

Step1: Isolate the rational exponent term

Add 15 to both sides.
$$(x^2 - x - 29)^{3/4} = 12 + 15$$
$$(x^2 - x - 29)^{3/4} = 27$$

Step2: Eliminate the rational exponent

Raise both sides to the $\frac{4}{3}$ power.
$$x^2 - x - 29 = 27^{4/3}$$
$$x^2 - x - 29 = (27^{1/3})^4 = 3^4 = 81$$

Step3: Rearrange to quadratic equation

Subtract 81 from both sides.
$$x^2 - x - 29 - 81 = 0$$
$$x^2 - x - 110 = 0$$

Step4: Factor the quadratic equation

Find two factors of -110 that sum to -1.
$$(x - 11)(x + 10) = 0$$

Step5: Solve for x

Set each factor equal to 0 and verify solutions (ensure the original expression under the rational exponent is non-negative, as we have a 4th root):
For $x=11$: $11^2 - 11 - 29 = 121 - 11 - 29 = 81 \geq 0$ (valid)
For $x=-10$: $(-10)^2 - (-10) - 29 = 100 + 10 - 29 = 81 \geq 0$ (valid)

Answer:

A. The solution(s) is(are) $x=11, -10$