Solve for m cube root of 3m+1=4

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

To remove the radical on the left side of the equation, cube both sides of the equation.

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

Simplify each side of the equation.

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

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

Simplify the left side.

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

$3 m + 1 = 64$

Solve for $m$ .

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

Subtract $1$ from both sides of the equation.

$3 m = 64 - 1$

Subtract $1$ from $64$ .

$3 m = 63$

Divide each term in $3 m = 63$ by $3$ and simplify.

Divide each term in $3 m = 63$ by $3$ .

$\frac{3 m}{3} = \frac{63}{3}$

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

$\frac{3 m}{3} = \frac{63}{3}$

Divide $m$ by $1$ .

$m = \frac{63}{3}$

Simplify the right side.

Divide $63$ by $3$ .

$m = 21$