Solve by Factoring (x-1)^(2/3)=100

${(x - 1)}^{\frac{2}{3}} = 100$

Step 1

Subtract $100$ from both sides of the equation.

${(x - 1)}^{\frac{2}{3}} - 100 = 0$

Step 2

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

${({(x - 1)}^{\frac{1}{3}})}^{2} - 100 = 0$

Step 3

Rewrite $100$ as $10^{2}$ .

${({(x - 1)}^{\frac{1}{3}})}^{2} - 10^{2} = 0$

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 $b = 10$ .

$({(x - 1)}^{\frac{1}{3}} + 10) ({(x - 1)}^{\frac{1}{3}} - 10) = 0$

Step 5

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

${(x - 1)}^{\frac{1}{3}} + 10 = 0$

${(x - 1)}^{\frac{1}{3}} - 10 = 0$

Step 6

Set ${(x - 1)}^{\frac{1}{3}} + 10$ equal to $0$ and solve for $x$ .

Set ${(x - 1)}^{\frac{1}{3}} + 10$ equal to $0$ .

${(x - 1)}^{\frac{1}{3}} + 10 = 0$

Solve ${(x - 1)}^{\frac{1}{3}} + 10 = 0$ for $x$ .

Subtract $10$ from both sides of the equation.

${(x - 1)}^{\frac{1}{3}} = - 10$

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

${({(x - 1)}^{\frac{1}{3}})}^{3} = {(- 10)}^{3}$

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

${(x - 1)}^{\frac{1}{3} \cdot 3} = {(- 10)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x - 1)}^{\frac{1}{3} \cdot 3} = {(- 10)}^{3}$

Rewrite the expression.

${(x - 1)}^{1} = {(- 10)}^{3}$

Simplify.

$x - 1 = {(- 10)}^{3}$

Simplify the right side.

Raise $- 10$ to the power of $3$ .

$x - 1 = - 1000$

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

Add $1$ to both sides of the equation.

$x = - 1000 + 1$

Add $- 1000$ and $1$ .

$x = - 999$

Step 7

Set ${(x - 1)}^{\frac{1}{3}} - 10$ equal to $0$ and solve for $x$ .

Set ${(x - 1)}^{\frac{1}{3}} - 10$ equal to $0$ .

${(x - 1)}^{\frac{1}{3}} - 10 = 0$

Solve ${(x - 1)}^{\frac{1}{3}} - 10 = 0$ for $x$ .

Add $10$ to both sides of the equation.

${(x - 1)}^{\frac{1}{3}} = 10$

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

${({(x - 1)}^{\frac{1}{3}})}^{3} = 10^{3}$

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

${(x - 1)}^{\frac{1}{3} \cdot 3} = 10^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

${(x - 1)}^{\frac{1}{3} \cdot 3} = 10^{3}$

Rewrite the expression.

${(x - 1)}^{1} = 10^{3}$

Simplify.

$x - 1 = 10^{3}$

Simplify the right side.

Raise $10$ to the power of $3$ .

$x - 1 = 1000$

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

Add $1$ to both sides of the equation.

$x = 1000 + 1$

Add $1000$ and $1$ .

$x = 1001$

Step 8

The final solution is all the values that make $({(x - 1)}^{\frac{1}{3}} + 10) ({(x - 1)}^{\frac{1}{3}} - 10) = 0$ true.

$x = - 999, 1001$

Solve by Factoring (x-1)^(2/3)=100

Related Questions

Support LCMGCF.com by disabling your adblocker.