Solve for x (x-1)^(2/3)=(x+5)^(1/3)

${(x - 1)}^{\frac{2}{3}} = {(x + 5)}^{\frac{1}{3}}$

Step 1

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

${({(x - 1)}^{\frac{2}{3}})}^{3} = {({(x + 5)}^{\frac{1}{3}})}^{3}$

Step 2

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

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

${(x - 1)}^{\frac{2}{3} \cdot 3} = {({(x + 5)}^{\frac{1}{3}})}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x - 1)}^{\frac{2}{3} \cdot 3} = {({(x + 5)}^{\frac{1}{3}})}^{3}$

Rewrite the expression.

${(x - 1)}^{2} = {({(x + 5)}^{\frac{1}{3}})}^{3}$

Step 3

Simplify ${({(x + 5)}^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${({(x + 5)}^{\frac{1}{3}})}^{3}$ .

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

${(x - 1)}^{2} = {(x + 5)}^{\frac{1}{3} \cdot 3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x - 1)}^{2} = {(x + 5)}^{\frac{1}{3} \cdot 3}$

Rewrite the expression.

${(x - 1)}^{2} = {(x + 5)}^{1}$

Simplify.

${(x - 1)}^{2} = x + 5$

Step 4

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

Subtract $x$ from both sides of the equation.

${(x - 1)}^{2} - x = 5$

Simplify each term.

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

$(x - 1) (x - 1) - x = 5$

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

Apply the distributive property.

$x (x - 1) - 1 (x - 1) - x = 5$

Apply the distributive property.

$x \cdot x + x \cdot - 1 - 1 (x - 1) - x = 5$

Apply the distributive property.

$x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1 - x = 5$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1 - x = 5$

Move $- 1$ to the left of $x$ .

$x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1 - x = 5$

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x - 1 x - 1 \cdot - 1 - x = 5$

Rewrite $- 1 x$ as $- x$ .

$x^{2} - x - x - 1 \cdot - 1 - x = 5$

Multiply $- 1$ by $- 1$ .

$x^{2} - x - x + 1 - x = 5$

Subtract $x$ from $- x$ .

$x^{2} - 2 x + 1 - x = 5$

Subtract $x$ from $- 2 x$ .

$x^{2} - 3 x + 1 = 5$

Step 5

Subtract $5$ from both sides of the equation.

$x^{2} - 3 x + 1 - 5 = 0$

Step 6

Subtract $5$ from $1$ .

$x^{2} - 3 x - 4 = 0$

Step 7

Factor $x^{2} - 3 x - 4$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 4$ and whose sum is $- 3$ .

$- 4, 1$

Write the factored form using these integers.

$(x - 4) (x + 1) = 0$

Step 8

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

$x - 4 = 0$

$x + 1 = 0$

Step 9

Set $x - 4$ equal to $0$ and solve for $x$ .

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Add $4$ to both sides of the equation.

$x = 4$

Step 10

Set $x + 1$ equal to $0$ and solve for $x$ .

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 11

The final solution is all the values that make $(x - 4) (x + 1) = 0$ true.

$x = 4, - 1$

Solve for x (x-1)^(2/3)=(x+5)^(1/3)

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