Solve by Factoring (x-1)^(2/3)=16

${(x - 1)}^{\frac{2}{3}} = 16$

Step 1

Subtract $16$ from both sides of the equation.

${(x - 1)}^{\frac{2}{3}} - 16 = 0$

Step 2

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

${({(x - 1)}^{\frac{1}{3}})}^{2} - 16 = 0$

Step 3

Rewrite $16$ as $4^{2}$ .

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

Step 4

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

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

Step 5

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

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

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

Step 6

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

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

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

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

Subtract $4$ from both sides of the equation.

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

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

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

Simplify the exponent.

Simplify the left side.

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Raise $- 4$ to the power of $3$ .

$x - 1 = - 64$

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

Add $1$ to both sides of the equation.

$x = - 64 + 1$

Add $- 64$ and $1$ .

$x = - 63$

Step 7

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

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

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

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

Add $4$ to both sides of the equation.

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

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

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

Simplify the exponent.

Simplify the left side.

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

$x - 1 = 4^{3}$

Simplify the right side.

Raise $4$ to the power of $3$ .

$x - 1 = 64$

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

Add $1$ to both sides of the equation.

$x = 64 + 1$

Add $64$ and $1$ .

$x = 65$

Step 8

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

$x = - 63, 65$

Solve by Factoring (x-1)^(2/3)=16

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