Solve by Factoring (x-3)^(2/3)=9

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

Step 1

Subtract $9$ from both sides of the equation.

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

Step 2

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

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

Step 3

Rewrite $9$ as $3^{2}$ .

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

Step 4

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

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

Step 5

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

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

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

Step 6

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

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

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

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

Subtract $3$ from both sides of the equation.

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

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

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

Simplify the exponent.

Simplify the left side.

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

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

$x - 3 = - 27$

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

Add $3$ to both sides of the equation.

$x = - 27 + 3$

Add $- 27$ and $3$ .

$x = - 24$

Step 7

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

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

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

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

Add $3$ to both sides of the equation.

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

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

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

Simplify the exponent.

Simplify the left side.

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$x - 3 = 3^{3}$

Simplify the right side.

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

$x - 3 = 27$

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

Add $3$ to both sides of the equation.

$x = 27 + 3$

Add $27$ and $3$ .

$x = 30$

Step 8

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

$x = - 24, 30$

Solve by Factoring (x-3)^(2/3)=9

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