Solve by Factoring x^(-2/3)+x^(-1/3)-2=0

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

Step 1

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

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

Step 2

Let $u = x^{- \frac{1}{3}}$ . Substitute $u$ for all occurrences of $x^{- \frac{1}{3}}$ .

$u^{2} + u - 2 = 0$

Step 3

Factor $u^{2} + u - 2$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 2$ and whose sum is $1$ .

$- 1, 2$

Write the factored form using these integers.

$(u - 1) (u + 2) = 0$

Step 4

Replace all occurrences of $u$ with $x^{- \frac{1}{3}}$ .

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

Step 5

Simplify.

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 6

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ .

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

Step 7

Combine $- 1$ and $\frac{x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ .

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

To write $2$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ .

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

Step 10

Combine the numerators over the common denominator.

$\frac{1 - x^{\frac{1}{3}}}{x^{\frac{1}{3}}} \cdot \frac{1 + 2 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} = 0$

Step 11

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

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

$\frac{1 + 2 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} = 0$

Step 12

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

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

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

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

Set the numerator equal to zero.

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

Solve the equation for $x$ .

Subtract $1$ from both sides of the equation.

$- x^{\frac{1}{3}} = - 1$

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

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

Simplify the exponent.

Simplify the left side.

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

Apply the product rule to $- x^{\frac{1}{3}}$ .

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

Raise $- 1$ to the power of $3$ .

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Raise $- 1$ to the power of $3$ .

$- x = - 1$

Divide each term in $- x = - 1$ by $- 1$ and simplify.

Divide each term in $- x = - 1$ by $- 1$ .

$\frac{- x}{- 1} = \frac{- 1}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{- 1}{- 1}$

Divide $x$ by $1$ .

$x = \frac{- 1}{- 1}$

Simplify the right side.

Divide $- 1$ by $- 1$ .

$x = 1$

Step 13

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

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

$\frac{1 + 2 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} = 0$

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

Set the numerator equal to zero.

$1 + 2 x^{\frac{1}{3}} = 0$

Solve the equation for $x$ .

Subtract $1$ from both sides of the equation.

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

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

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

Simplify the exponent.

Simplify the left side.

Simplify ${(2 x^{\frac{1}{3}})}^{3}$ .

Apply the product rule to $2 x^{\frac{1}{3}}$ .

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

Raise $2$ to the power of $3$ .

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

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Simplify.

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

Simplify the right side.

Raise $- 1$ to the power of $3$ .

$8 x = - 1$

Divide each term in $8 x = - 1$ by $8$ and simplify.

Divide each term in $8 x = - 1$ by $8$ .

$\frac{8 x}{8} = \frac{- 1}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x}{8} = \frac{- 1}{8}$

Divide $x$ by $1$ .

$x = \frac{- 1}{8}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{8}$

Step 14

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

$x = 1, - \frac{1}{8}$

Solve by Factoring x^(-2/3)+x^(-1/3)-2=0

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