Solve for r cube root of 8r+1 = cube root of r+8

$\sqrt[3]{8 r + 1} = \sqrt[3]{r + 8}$

Step 1

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

${\sqrt[3]{8 r + 1}}^{3} = {\sqrt[3]{r + 8}}^{3}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{8 r + 1}$ as ${(8 r + 1)}^{\frac{1}{3}}$ .

${({(8 r + 1)}^{\frac{1}{3}})}^{3} = {\sqrt[3]{r + 8}}^{3}$

Simplify the left side.

Simplify ${({(8 r + 1)}^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${({(8 r + 1)}^{\frac{1}{3}})}^{3}$ .

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

${(8 r + 1)}^{\frac{1}{3} \cdot 3} = {\sqrt[3]{r + 8}}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(8 r + 1)}^{\frac{1}{3} \cdot 3} = {\sqrt[3]{r + 8}}^{3}$

Rewrite the expression.

${(8 r + 1)}^{1} = {\sqrt[3]{r + 8}}^{3}$

Simplify.

$8 r + 1 = {\sqrt[3]{r + 8}}^{3}$

Simplify the right side.

Rewrite ${\sqrt[3]{r + 8}}^{3}$ as $r + 8$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{r + 8}$ as ${(r + 8)}^{\frac{1}{3}}$ .

$8 r + 1 = {({(r + 8)}^{\frac{1}{3}})}^{3}$

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

$8 r + 1 = {(r + 8)}^{\frac{1}{3} \cdot 3}$

Combine $\frac{1}{3}$ and $3$ .

$8 r + 1 = {(r + 8)}^{\frac{3}{3}}$

Cancel the common factor of $3$ .

Cancel the common factor.

$8 r + 1 = {(r + 8)}^{\frac{3}{3}}$

Rewrite the expression.

$8 r + 1 = {(r + 8)}^{1}$

Simplify.

$8 r + 1 = r + 8$

Step 3

Solve for $r$ .

Move all terms containing $r$ to the left side of the equation.

Subtract $r$ from both sides of the equation.

$8 r + 1 - r = 8$

Subtract $r$ from $8 r$ .

$7 r + 1 = 8$

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

Subtract $1$ from both sides of the equation.

$7 r = 8 - 1$

Subtract $1$ from $8$ .

$7 r = 7$

Divide each term in $7 r = 7$ by $7$ and simplify.

Divide each term in $7 r = 7$ by $7$ .

$\frac{7 r}{7} = \frac{7}{7}$

Simplify the left side.

Cancel the common factor of $7$ .

Cancel the common factor.

$\frac{7 r}{7} = \frac{7}{7}$

Divide $r$ by $1$ .

$r = \frac{7}{7}$

Simplify the right side.

Divide $7$ by $7$ .

$r = 1$

Solve for r cube root of 8r+1 = cube root of r+8

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