Question

Solve:${\left(9x+9\right)}^{\frac{1}{4}}-2=1$

Short answer

The solution is,$x=8$

Step-by-step solution

Step 1 Given equation is,

${\left(9x+9\right)}^{\frac{1}{4}}-2=1$.

The objective is, we need to find the solution for the given equation.

Step 2 To isolate the radical add 2 on both sides.

$$\begin{gathered} {\left(9x+9\right)}^{\frac{1}{4}}-2+2=1+2 \\ {\left(9x+9\right)}^{\frac{1}{4}}=3 \end{gathered}$$

Raise each side of the equation to the fourth power.

${\left({\left(9x+9\right)}^{\frac{1}{4}}\right)}^4={\left(3\right)}^4$

Simplify it.

$$\begin{gathered} {\left({\left(9x+9\right)}^{\frac{1}{4}}\right)}^4={\left(3\right)}^4 \\ 9x+9=81 \\ 9x=81-9 \\ 9x=72 \\ x=\frac{72}{9} \\ x=8 \end{gathered}$$

Step 3 Let us check the answer.

Substitute $x=8$into the equation ${\left(9x+9\right)}^{\frac{1}{4}}-2=1$.

$$\begin{gathered} {\left(9\left(8\right)+9\right)}^{\frac{1}{4}}-2=1 \\ {\left(72+9\right)}^{\frac{1}{4}}-2=1 \\ {\left(81\right)}^{\frac{1}{4}}-2=1 \\ {\left(3^4\right)}^{\frac{1}{4}}-2=1 \\ 3-2=1 \\ 1=1 \end{gathered}$$

which is true.