Solve by Factoring (x-4)^(2/3)=64

${(x - 4)}^{\frac{2}{3}} = 64$

Step 1

Subtract $64$ from both sides of the equation.

${(x - 4)}^{\frac{2}{3}} - 64 = 0$

Step 2

Rewrite ${(x - 4)}^{\frac{2}{3}}$ as ${({(x - 4)}^{\frac{1}{3}})}^{2}$ .

${({(x - 4)}^{\frac{1}{3}})}^{2} - 64 = 0$

Step 3

Rewrite $64$ as $8^{2}$ .

${({(x - 4)}^{\frac{1}{3}})}^{2} - 8^{2} = 0$

Step 4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = {(x - 4)}^{\frac{1}{3}}$ and $b = 8$ .

$({(x - 4)}^{\frac{1}{3}} + 8) ({(x - 4)}^{\frac{1}{3}} - 8) = 0$

Step 5

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

${(x - 4)}^{\frac{1}{3}} + 8 = 0$

${(x - 4)}^{\frac{1}{3}} - 8 = 0$

Step 6

Set ${(x - 4)}^{\frac{1}{3}} + 8$ equal to $0$ and solve for $x$ .

Set ${(x - 4)}^{\frac{1}{3}} + 8$ equal to $0$ .

${(x - 4)}^{\frac{1}{3}} + 8 = 0$

Solve ${(x - 4)}^{\frac{1}{3}} + 8 = 0$ for $x$ .

Subtract $8$ from both sides of the equation.

${(x - 4)}^{\frac{1}{3}} = - 8$

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

${({(x - 4)}^{\frac{1}{3}})}^{3} = {(- 8)}^{3}$

Simplify the exponent.

Simplify the left side.

Simplify ${({(x - 4)}^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${({(x - 4)}^{\frac{1}{3}})}^{3}$ .

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

${(x - 4)}^{\frac{1}{3} \cdot 3} = {(- 8)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x - 4)}^{\frac{1}{3} \cdot 3} = {(- 8)}^{3}$

Rewrite the expression.

${(x - 4)}^{1} = {(- 8)}^{3}$

Simplify.

$x - 4 = {(- 8)}^{3}$

Simplify the right side.

Raise $- 8$ to the power of $3$ .

$x - 4 = - 512$

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

Add $4$ to both sides of the equation.

$x = - 512 + 4$

Add $- 512$ and $4$ .

$x = - 508$

Step 7

Set ${(x - 4)}^{\frac{1}{3}} - 8$ equal to $0$ and solve for $x$ .

Set ${(x - 4)}^{\frac{1}{3}} - 8$ equal to $0$ .

${(x - 4)}^{\frac{1}{3}} - 8 = 0$

Solve ${(x - 4)}^{\frac{1}{3}} - 8 = 0$ for $x$ .

Add $8$ to both sides of the equation.

${(x - 4)}^{\frac{1}{3}} = 8$

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

${({(x - 4)}^{\frac{1}{3}})}^{3} = 8^{3}$

Simplify the exponent.

Simplify the left side.

Simplify ${({(x - 4)}^{\frac{1}{3}})}^{3}$ .

Multiply the exponents in ${({(x - 4)}^{\frac{1}{3}})}^{3}$ .

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

${(x - 4)}^{\frac{1}{3} \cdot 3} = 8^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x - 4)}^{\frac{1}{3} \cdot 3} = 8^{3}$

Rewrite the expression.

${(x - 4)}^{1} = 8^{3}$

Simplify.

$x - 4 = 8^{3}$

Simplify the right side.

Raise $8$ to the power of $3$ .

$x - 4 = 512$

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

Add $4$ to both sides of the equation.

$x = 512 + 4$

Add $512$ and $4$ .

$x = 516$

Step 8

The final solution is all the values that make $({(x - 4)}^{\frac{1}{3}} + 8) ({(x - 4)}^{\frac{1}{3}} - 8) = 0$ true.

$x = - 508, 516$

Solve by Factoring (x-4)^(2/3)=64

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