Solve by Factoring (x-2)^(2/3)=49

${(x - 2)}^{\frac{2}{3}} = 49$

Step 1

Subtract $49$ from both sides of the equation.

${(x - 2)}^{\frac{2}{3}} - 49 = 0$

Step 2

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

${({(x - 2)}^{\frac{1}{3}})}^{2} - 49 = 0$

Step 3

Rewrite $49$ as $7^{2}$ .

${({(x - 2)}^{\frac{1}{3}})}^{2} - 7^{2} = 0$

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 $b = 7$ .

$({(x - 2)}^{\frac{1}{3}} + 7) ({(x - 2)}^{\frac{1}{3}} - 7) = 0$

Step 5

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

${(x - 2)}^{\frac{1}{3}} + 7 = 0$

${(x - 2)}^{\frac{1}{3}} - 7 = 0$

Step 6

Set ${(x - 2)}^{\frac{1}{3}} + 7$ equal to $0$ and solve for $x$ .

Set ${(x - 2)}^{\frac{1}{3}} + 7$ equal to $0$ .

${(x - 2)}^{\frac{1}{3}} + 7 = 0$

Solve ${(x - 2)}^{\frac{1}{3}} + 7 = 0$ for $x$ .

Subtract $7$ from both sides of the equation.

${(x - 2)}^{\frac{1}{3}} = - 7$

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

${({(x - 2)}^{\frac{1}{3}})}^{3} = {(- 7)}^{3}$

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

${(x - 2)}^{\frac{1}{3} \cdot 3} = {(- 7)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x - 2)}^{\frac{1}{3} \cdot 3} = {(- 7)}^{3}$

Rewrite the expression.

${(x - 2)}^{1} = {(- 7)}^{3}$

Simplify.

$x - 2 = {(- 7)}^{3}$

Simplify the right side.

Raise $- 7$ to the power of $3$ .

$x - 2 = - 343$

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

Add $2$ to both sides of the equation.

$x = - 343 + 2$

Add $- 343$ and $2$ .

$x = - 341$

Step 7

Set ${(x - 2)}^{\frac{1}{3}} - 7$ equal to $0$ and solve for $x$ .

Set ${(x - 2)}^{\frac{1}{3}} - 7$ equal to $0$ .

${(x - 2)}^{\frac{1}{3}} - 7 = 0$

Solve ${(x - 2)}^{\frac{1}{3}} - 7 = 0$ for $x$ .

Add $7$ to both sides of the equation.

${(x - 2)}^{\frac{1}{3}} = 7$

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

${({(x - 2)}^{\frac{1}{3}})}^{3} = 7^{3}$

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

${(x - 2)}^{\frac{1}{3} \cdot 3} = 7^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x - 2)}^{\frac{1}{3} \cdot 3} = 7^{3}$

Rewrite the expression.

${(x - 2)}^{1} = 7^{3}$

Simplify.

$x - 2 = 7^{3}$

Simplify the right side.

Raise $7$ to the power of $3$ .

$x - 2 = 343$

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

Add $2$ to both sides of the equation.

$x = 343 + 2$

Add $343$ and $2$ .

$x = 345$

Step 8

The final solution is all the values that make $({(x - 2)}^{\frac{1}{3}} + 7) ({(x - 2)}^{\frac{1}{3}} - 7) = 0$ true.

$x = - 341, 345$

Solve by Factoring (x-2)^(2/3)=49

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