Solve Equations with Rational Exponents
Solve and check the equation.
-$$\left( x ^ { 2 } + 6 x + 9 \right) ^ { 3 / 4 } - 10 = 17$$
A) $\{ - 12,6 \}$
B) $\{ 6 \}$
C) $\{ - 12,0,6 \}$
D) $\{ 27 \}$

Question

Quizplus · Accepted Answer

First, isolate the expression with the rational exponent by adding 10 to both sides, resulting in $(x^2 + 6x + 9)^{3/4} = 27$. Recognizing that $27 = 3^3$, we can raise both sides to the $4/3$ power to eliminate the exponent on the left, yielding $x^2 + 6x + 9 = 3^4 = 81$. Simplifying, we get $x^2 + 6x - 72 = 0$, which factors to $(x + 12)(x - 6) = 0$, giving the solutions $x = -12$ and $x = 6$.

Solve Equations with Rational Exponents Solve and Check the Equation (x2+6x+9)3/4−10=17\left( x ^ { 2 } + 6 x + 9 \right) ^ { 3 / 4 } - 10 = 17(x2+6x+9)3/4−10=17

Solve Equations with Rational Exponents
Solve and Check the Equation $\left( x ^ { 2 } + 6 x + 9 \right) ^ { 3 / 4 } - 10 = 17$