Solve by Factoring (x-2)^(2/3)=25

${(x - 2)}^{\frac{2}{3}} = 25$

Step 1

Subtract $25$ from both sides of the equation.

${(x - 2)}^{\frac{2}{3}} - 25 = 0$

Step 2

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

${({(x - 2)}^{\frac{1}{3}})}^{2} - 25 = 0$

Step 3

Rewrite $25$ as $5^{2}$ .

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

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {(x - 2)}^{\frac{1}{3}}$ and $b = 5$ .

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

Step 5

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

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

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

Step 6

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 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 $x$ .

Add $5$ to 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 $2$ .

$x = 127$

Step 8

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

$x = - 123, 127$

Solve by Factoring (x-2)^(2/3)=25

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