Find All Complex Solutions (x-1)^(2/5)=4

${(x - 1)}^{\frac{2}{5}} = 4$

Step 1

Subtract $4$ from both sides of the equation.

${(x - 1)}^{\frac{2}{5}} - 4 = 0$

Step 2

Factor the left side of the equation.

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

${({(x - 1)}^{\frac{1}{5}})}^{2} - 4 = 0$

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

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

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

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

Step 3

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

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

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

Step 4

Set ${(x - 1)}^{\frac{1}{5}} + 2$ equal to $0$ and solve for $x$ .

Set ${(x - 1)}^{\frac{1}{5}} + 2$ equal to $0$ .

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

Solve ${(x - 1)}^{\frac{1}{5}} + 2 = 0$ for $x$ .

Subtract $2$ from both sides of the equation.

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

Raise each side of the equation to the power of $5$ to eliminate the fractional exponent on the left side.

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

Simplify the exponent.

Simplify the left side.

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

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

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

${(x - 1)}^{\frac{1}{5} \cdot 5} = {(- 2)}^{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

${(x - 1)}^{\frac{1}{5} \cdot 5} = {(- 2)}^{5}$

Rewrite the expression.

${(x - 1)}^{1} = {(- 2)}^{5}$

Simplify.

$x - 1 = {(- 2)}^{5}$

Simplify the right side.

Raise $- 2$ to the power of $5$ .

$x - 1 = - 32$

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

Add $1$ to both sides of the equation.

$x = - 32 + 1$

Add $- 32$ and $1$ .

$x = - 31$

Step 5

Set ${(x - 1)}^{\frac{1}{5}} - 2$ equal to $0$ and solve for $x$ .

Set ${(x - 1)}^{\frac{1}{5}} - 2$ equal to $0$ .

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

Solve ${(x - 1)}^{\frac{1}{5}} - 2 = 0$ for $x$ .

Add $2$ to both sides of the equation.

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

Raise each side of the equation to the power of $5$ to eliminate the fractional exponent on the left side.

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

Simplify the exponent.

Simplify the left side.

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

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

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

${(x - 1)}^{\frac{1}{5} \cdot 5} = 2^{5}$

Cancel the common factor of $5$ .

Cancel the common factor.

${(x - 1)}^{\frac{1}{5} \cdot 5} = 2^{5}$

Rewrite the expression.

${(x - 1)}^{1} = 2^{5}$

Simplify.

$x - 1 = 2^{5}$

Simplify the right side.

Raise $2$ to the power of $5$ .

$x - 1 = 32$

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

Add $1$ to both sides of the equation.

$x = 32 + 1$

Add $32$ and $1$ .

$x = 33$

Step 6

The final solution is all the values that make $({(x - 1)}^{\frac{1}{5}} + 2) ({(x - 1)}^{\frac{1}{5}} - 2) = 0$ true.

$x = - 31, 33$

Find All Complex Solutions (x-1)^(2/5)=4

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