Solve by Factoring (x-5)^(2/3)=81

${(x - 5)}^{\frac{2}{3}} = 81$

Step 1

Subtract $81$ from both sides of the equation.

${(x - 5)}^{\frac{2}{3}} - 81 = 0$

Step 2

Rewrite ${(x - 5)}^{\frac{2}{3}}$ as ${({(x - 5)}^{\frac{1}{3}})}^{2}$ .

${({(x - 5)}^{\frac{1}{3}})}^{2} - 81 = 0$

Step 3

Rewrite $81$ as $9^{2}$ .

${({(x - 5)}^{\frac{1}{3}})}^{2} - 9^{2} = 0$

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {(x - 5)}^{\frac{1}{3}}$ and $b = 9$ .

$({(x - 5)}^{\frac{1}{3}} + 9) ({(x - 5)}^{\frac{1}{3}} - 9) = 0$

Step 5

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

${(x - 5)}^{\frac{1}{3}} + 9 = 0$

${(x - 5)}^{\frac{1}{3}} - 9 = 0$

Step 6

Set ${(x - 5)}^{\frac{1}{3}} + 9$ equal to $0$ and solve for $x$ .

Set ${(x - 5)}^{\frac{1}{3}} + 9$ equal to $0$ .

${(x - 5)}^{\frac{1}{3}} + 9 = 0$

Solve ${(x - 5)}^{\frac{1}{3}} + 9 = 0$ for $x$ .

Subtract $9$ from both sides of the equation.

${(x - 5)}^{\frac{1}{3}} = - 9$

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

${({(x - 5)}^{\frac{1}{3}})}^{3} = {(- 9)}^{3}$

Simplify the exponent.

Simplify the left side.

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

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

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

${(x - 5)}^{\frac{1}{3} \cdot 3} = {(- 9)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x - 5)}^{\frac{1}{3} \cdot 3} = {(- 9)}^{3}$

Rewrite the expression.

${(x - 5)}^{1} = {(- 9)}^{3}$

Simplify.

$x - 5 = {(- 9)}^{3}$

Simplify the right side.

Raise $- 9$ to the power of $3$ .

$x - 5 = - 729$

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

Add $5$ to both sides of the equation.

$x = - 729 + 5$

Add $- 729$ and $5$ .

$x = - 724$

Step 7

Set ${(x - 5)}^{\frac{1}{3}} - 9$ equal to $0$ and solve for $x$ .

Set ${(x - 5)}^{\frac{1}{3}} - 9$ equal to $0$ .

${(x - 5)}^{\frac{1}{3}} - 9 = 0$

Solve ${(x - 5)}^{\frac{1}{3}} - 9 = 0$ for $x$ .

Add $9$ to both sides of the equation.

${(x - 5)}^{\frac{1}{3}} = 9$

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

${({(x - 5)}^{\frac{1}{3}})}^{3} = 9^{3}$

Simplify the exponent.

Simplify the left side.

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

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

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

${(x - 5)}^{\frac{1}{3} \cdot 3} = 9^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x - 5)}^{\frac{1}{3} \cdot 3} = 9^{3}$

Rewrite the expression.

${(x - 5)}^{1} = 9^{3}$

Simplify.

$x - 5 = 9^{3}$

Simplify the right side.

Raise $9$ to the power of $3$ .

$x - 5 = 729$

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

Add $5$ to both sides of the equation.

$x = 729 + 5$

Add $729$ and $5$ .

$x = 734$

Step 8

The final solution is all the values that make $({(x - 5)}^{\frac{1}{3}} + 9) ({(x - 5)}^{\frac{1}{3}} - 9) = 0$ true.

$x = - 724, 734$

Solve by Factoring (x-5)^(2/3)=81

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