Solve for x (x+1)^(2/3)=(9x+1)^(1/3)

${(x + 1)}^{\frac{2}{3}} = {(9 x + 1)}^{\frac{1}{3}}$

Step 1

Eliminate the fractional exponents by multiplying both exponents by the LCD.

${({(x + 1)}^{\frac{2}{3}})}^{3} = {({(9 x + 1)}^{\frac{1}{3}})}^{3}$

Step 2

Multiply the exponents in ${({(x + 1)}^{\frac{2}{3}})}^{3}$ .

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

${(x + 1)}^{\frac{2}{3} \cdot 3} = {({(9 x + 1)}^{\frac{1}{3}})}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x + 1)}^{\frac{2}{3} \cdot 3} = {({(9 x + 1)}^{\frac{1}{3}})}^{3}$

Rewrite the expression.

${(x + 1)}^{2} = {({(9 x + 1)}^{\frac{1}{3}})}^{3}$

Step 3

Simplify ${({(9 x + 1)}^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${({(9 x + 1)}^{\frac{1}{3}})}^{3}$ .

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

${(x + 1)}^{2} = {(9 x + 1)}^{\frac{1}{3} \cdot 3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x + 1)}^{2} = {(9 x + 1)}^{\frac{1}{3} \cdot 3}$

Rewrite the expression.

${(x + 1)}^{2} = {(9 x + 1)}^{1}$

Simplify.

${(x + 1)}^{2} = 9 x + 1$

Step 4

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

Subtract $9 x$ from both sides of the equation.

${(x + 1)}^{2} - 9 x = 1$

Simplify each term.

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$(x + 1) (x + 1) - 9 x = 1$

Expand $(x + 1) (x + 1)$ using the FOIL Method.

Apply the distributive property.

$x (x + 1) + 1 (x + 1) - 9 x = 1$

Apply the distributive property.

$x \cdot x + x \cdot 1 + 1 (x + 1) - 9 x = 1$

Apply the distributive property.

$x \cdot x + x \cdot 1 + 1 x + 1 \cdot 1 - 9 x = 1$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot 1 + 1 x + 1 \cdot 1 - 9 x = 1$

Multiply $x$ by $1$ .

$x^{2} + x + 1 x + 1 \cdot 1 - 9 x = 1$

Multiply $x$ by $1$ .

$x^{2} + x + x + 1 \cdot 1 - 9 x = 1$

Multiply $1$ by $1$ .

$x^{2} + x + x + 1 - 9 x = 1$

Add $x$ and $x$ .

$x^{2} + 2 x + 1 - 9 x = 1$

Subtract $9 x$ from $2 x$ .

$x^{2} - 7 x + 1 = 1$

Step 5

Subtract $1$ from both sides of the equation.

$x^{2} - 7 x + 1 - 1 = 0$

Step 6

Combine the opposite terms in $x^{2} - 7 x + 1 - 1$ .

Subtract $1$ from $1$ .

$x^{2} - 7 x + 0 = 0$

Add $x^{2} - 7 x$ and $0$ .

$x^{2} - 7 x = 0$

Step 7

Factor $x$ out of $x^{2} - 7 x$ .

Factor $x$ out of $x^{2}$ .

$x \cdot x - 7 x = 0$

Factor $x$ out of $- 7 x$ .

$x \cdot x + x \cdot - 7 = 0$

Factor $x$ out of $x \cdot x + x \cdot - 7$ .

$x (x - 7) = 0$

Step 8

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x - 7 = 0$

Step 9

Set $x$ equal to $0$ .

$x = 0$

Step 10

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

Step 11

The final solution is all the values that make $x (x - 7) = 0$ true.

$x = 0, 7$

Solve for x (x+1)^(2/3)=(9x+1)^(1/3)

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