Solve by Factoring fourth root of 3x+1-3=0

$\sqrt[4]{3 x + 1} - 3 = 0$

Step 1

Add $3$ to both sides of the equation.

$\sqrt[4]{3 x + 1} = 3$

Step 2

To remove the radical on the left side of the equation, raise both sides of the equation to the power of $4$ .

${\sqrt[4]{3 x + 1}}^{4} = 3^{4}$

Step 3

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{3 x + 1}$ as ${(3 x + 1)}^{\frac{1}{4}}$ .

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

Simplify the left side.

Simplify ${({(3 x + 1)}^{\frac{1}{4}})}^{4}$ .

Multiply the exponents in ${({(3 x + 1)}^{\frac{1}{4}})}^{4}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(3 x + 1)}^{\frac{1}{4} \cdot 4} = 3^{4}$

Cancel the common factor of $4$ .

Cancel the common factor.

${(3 x + 1)}^{\frac{1}{4} \cdot 4} = 3^{4}$

Rewrite the expression.

${(3 x + 1)}^{1} = 3^{4}$

Simplify.

$3 x + 1 = 3^{4}$

Simplify the right side.

Raise $3$ to the power of $4$ .

$3 x + 1 = 81$

Step 4

Solve for $x$ .

Move all terms not containing $x$ to the right side of the equation.

Subtract $1$ from both sides of the equation.

$3 x = 81 - 1$

Subtract $1$ from $81$ .

$3 x = 80$

Divide each term in $3 x = 80$ by $3$ and simplify.

Divide each term in $3 x = 80$ by $3$ .

$\frac{3 x}{3} = \frac{80}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 x}{3} = \frac{80}{3}$

Divide $x$ by $1$ .

$x = \frac{80}{3}$