Solve by Factoring (x-2)^(2/3)+(x-2)^(1/3)-20=0

${(x - 2)}^{\frac{2}{3}} + {(x - 2)}^{\frac{1}{3}} - 20 = 0$

Step 1

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

${({(x - 2)}^{\frac{1}{3}})}^{2} + {(x - 2)}^{\frac{1}{3}} - 20 = 0$

Step 2

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

$u^{2} + u - 20 = 0$

Step 3

Factor $u^{2} + u - 20$ 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 $- 20$ and whose sum is $1$ .

$- 4, 5$

Write the factored form using these integers.

$(u - 4) (u + 5) = 0$

Step 4

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

$({(x - 2)}^{\frac{1}{3}} - 4) ({(x - 2)}^{\frac{1}{3}} + 5) = 0$

Step 5

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

${(x - 2)}^{\frac{1}{3}} - 4 = 0$

${(x - 2)}^{\frac{1}{3}} + 5 = 0$

Step 6

Set ${(x - 2)}^{\frac{1}{3}} - 4$ equal to $0$ and solve for $x$ .

Set ${(x - 2)}^{\frac{1}{3}} - 4$ equal to $0$ .

${(x - 2)}^{\frac{1}{3}} - 4 = 0$

Solve ${(x - 2)}^{\frac{1}{3}} - 4 = 0$ for $x$ .

Add $4$ to both sides of the equation.

${(x - 2)}^{\frac{1}{3}} = 4$

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

${({(x - 2)}^{\frac{1}{3}})}^{3} = 4^{3}$

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, ${(a^{m})}^{n} = a^{m n}$ .

${(x - 2)}^{\frac{1}{3} \cdot 3} = 4^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x - 2)}^{\frac{1}{3} \cdot 3} = 4^{3}$

Rewrite the expression.

${(x - 2)}^{1} = 4^{3}$

Simplify.

$x - 2 = 4^{3}$

Simplify the right side.

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

$x - 2 = 64$

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

Add $2$ to both sides of the equation.

$x = 64 + 2$

Add $64$ and $2$ .

$x = 66$

Step 7

Set ${(x - 2)}^{\frac{1}{3}} + 5$ equal to $0$ and solve for $x$ .

Set ${(x - 2)}^{\frac{1}{3}} + 5$ equal to $0$ .

${(x - 2)}^{\frac{1}{3}} + 5 = 0$

Solve ${(x - 2)}^{\frac{1}{3}} + 5 = 0$ for $x$ .

Subtract $5$ from both sides of the equation.

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

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

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$x - 2 = {(- 5)}^{3}$

Simplify the right side.

Raise $- 5$ to the power of $3$ .

$x - 2 = - 125$

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

Add $2$ to both sides of the equation.

$x = - 125 + 2$

Add $- 125$ and $2$ .

$x = - 123$

Step 8

The final solution is all the values that make $({(x - 2)}^{\frac{1}{3}} - 4) ({(x - 2)}^{\frac{1}{3}} + 5) = 0$ true.

$x = 66, - 123$

Solve by Factoring (x-2)^(2/3)+(x-2)^(1/3)-20=0

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