Solve by Factoring (x-2)^(2/3)+(x-2)^(1/3)-2=0

${(x - 2)}^{\frac{2}{3}} + {(x - 2)}^{\frac{1}{3}} - 2 = 0$

Step 1

Rewrite ${(x - 2)}^{\frac{2}{3}}$ as ${({(x - 2)}^{\frac{1}{3}})}^{2}$ .

${({(x - 2)}^{\frac{1}{3}})}^{2} + {(x - 2)}^{\frac{1}{3}} - 2 = 0$

Step 2

Let $u = {(x - 2)}^{\frac{1}{3}}$ . Substitute $u$ for all occurrences of ${(x - 2)}^{\frac{1}{3}}$ .

$u^{2} + u - 2 = 0$

Step 3

Factor $u^{2} + u - 2$ 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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Write the factored form using these integers.

$(u - 1) (u + 2) = 0$

Step 4

Replace all occurrences of $u$ with ${(x - 2)}^{\frac{1}{3}}$ .

$({(x - 2)}^{\frac{1}{3}} - 1) ({(x - 2)}^{\frac{1}{3}} + 2) = 0$

Step 5

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

${(x - 2)}^{\frac{1}{3}} - 1 = 0$

${(x - 2)}^{\frac{1}{3}} + 2 = 0$

Step 6

Set ${(x - 2)}^{\frac{1}{3}} - 1$ equal to $0$ and solve for $x$ .

Set ${(x - 2)}^{\frac{1}{3}} - 1$ equal to $0$ .

${(x - 2)}^{\frac{1}{3}} - 1 = 0$

Solve ${(x - 2)}^{\frac{1}{3}} - 1 = 0$ for $x$ .

Add $1$ to both sides of the equation.

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

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

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

Simplify the exponent.

Simplify the left side.

Simplify ${({(x - 2)}^{\frac{1}{3}})}^{3}$ .

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

${(x - 2)}^{1} = 1^{3}$

Simplify.

$x - 2 = 1^{3}$

Simplify the right side.

One to any power is one.

$x - 2 = 1$

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

Add $2$ to both sides of the equation.

$x = 1 + 2$

Add $1$ and $2$ .

$x = 3$

Step 7

Set ${(x - 2)}^{\frac{1}{3}} + 2$ equal to $0$ and solve for $x$ .

Set ${(x - 2)}^{\frac{1}{3}} + 2$ equal to $0$ .

${(x - 2)}^{\frac{1}{3}} + 2 = 0$

Solve ${(x - 2)}^{\frac{1}{3}} + 2 = 0$ for $x$ .

Subtract $2$ from both sides of the equation.

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

Raise each side of the equation to the power of $3$ to eliminate the fractional exponent on the left side.

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

Simplify the exponent.

Simplify the left side.

Simplify ${({(x - 2)}^{\frac{1}{3}})}^{3}$ .

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

${(x - 2)}^{1} = {(- 2)}^{3}$

Simplify.

$x - 2 = {(- 2)}^{3}$

Simplify the right side.

Raise $- 2$ to the power of $3$ .

$x - 2 = - 8$

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

Add $2$ to both sides of the equation.

$x = - 8 + 2$

Add $- 8$ and $2$ .

$x = - 6$

Step 8

The final solution is all the values that make $({(x - 2)}^{\frac{1}{3}} - 1) ({(x - 2)}^{\frac{1}{3}} + 2) = 0$ true.

$x = 3, - 6$

Solve by Factoring (x-2)^(2/3)+(x-2)^(1/3)-2=0

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