Solve by Factoring x^(-2/3)+x^(-1/3)-56=0

$x^{- \frac{2}{3}} + x^{- \frac{1}{3}} - 56 = 0$

Step 1

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

${(x^{- \frac{1}{3}})}^{2} + x^{- \frac{1}{3}} - 56 = 0$

Step 2

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

$u^{2} + u - 56 = 0$

Step 3

Factor $u^{2} + u - 56$ 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 $- 56$ and whose sum is $1$ .

$- 7, 8$

Write the factored form using these integers.

$(u - 7) (u + 8) = 0$

Step 4

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

$(x^{- \frac{1}{3}} - 7) (x^{- \frac{1}{3}} + 8) = 0$

Step 5

Simplify.

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

$(\frac{1}{x^{\frac{1}{3}}} - 7) (x^{- \frac{1}{3}} + 8) = 0$

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

$(\frac{1}{x^{\frac{1}{3}}} - 7) (\frac{1}{x^{\frac{1}{3}}} + 8) = 0$

Step 6

To write $- 7$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ .

$(\frac{1}{x^{\frac{1}{3}}} - 7 \cdot \frac{x^{\frac{1}{3}}}{x^{\frac{1}{3}}}) (\frac{1}{x^{\frac{1}{3}}} + 8) = 0$

Step 7

Combine $- 7$ and $\frac{x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ .

$(\frac{1}{x^{\frac{1}{3}}} + \frac{- 7 x^{\frac{1}{3}}}{x^{\frac{1}{3}}}) (\frac{1}{x^{\frac{1}{3}}} + 8) = 0$

Step 8

Combine the numerators over the common denominator.

$\frac{1 - 7 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} \cdot (\frac{1}{x^{\frac{1}{3}}} + 8) = 0$

Step 9

To write $8$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ .

$\frac{1 - 7 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} \cdot (\frac{1}{x^{\frac{1}{3}}} + \frac{8 x^{\frac{1}{3}}}{x^{\frac{1}{3}}}) = 0$

Step 10

Combine the numerators over the common denominator.

$\frac{1 - 7 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} \cdot \frac{1 + 8 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} = 0$

Step 11

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

$\frac{1 - 7 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} = 0$

$\frac{1 + 8 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} = 0$

Step 12

Set $\frac{1 - 7 x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ equal to $0$ and solve for $x$ .

Set $\frac{1 - 7 x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ equal to $0$ .

$\frac{1 - 7 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} = 0$

Solve $\frac{1 - 7 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} = 0$ for $x$ .

Set the numerator equal to zero.

$1 - 7 x^{\frac{1}{3}} = 0$

Solve the equation for $x$ .

Subtract $1$ from both sides of the equation.

$- 7 x^{\frac{1}{3}} = - 1$

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

${(- 7 x^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Simplify the exponent.

Simplify the left side.

Simplify ${(- 7 x^{\frac{1}{3}})}^{3}$ .

Apply the product rule to $- 7 x^{\frac{1}{3}}$ .

${(- 7)}^{3} {(x^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Raise $- 7$ to the power of $3$ .

$- 343 {(x^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

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

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

$- 343 x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$- 343 x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Rewrite the expression.

$- 343 x^{1} = {(- 1)}^{3}$

Simplify.

$- 343 x = {(- 1)}^{3}$

Simplify the right side.

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

$- 343 x = - 1$

Divide each term in $- 343 x = - 1$ by $- 343$ and simplify.

Divide each term in $- 343 x = - 1$ by $- 343$ .

$\frac{- 343 x}{- 343} = \frac{- 1}{- 343}$

Simplify the left side.

Cancel the common factor of $- 343$ .

Cancel the common factor.

$\frac{- 343 x}{- 343} = \frac{- 1}{- 343}$

Divide $x$ by $1$ .

$x = \frac{- 1}{- 343}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{1}{343}$

Step 13

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

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

$\frac{1 + 8 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} = 0$

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

Set the numerator equal to zero.

$1 + 8 x^{\frac{1}{3}} = 0$

Solve the equation for $x$ .

Subtract $1$ from both sides of the equation.

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

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

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

Simplify the exponent.

Simplify the left side.

Simplify ${(8 x^{\frac{1}{3}})}^{3}$ .

Apply the product rule to $8 x^{\frac{1}{3}}$ .

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

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

$512 {(x^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

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

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

$512 x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Cancel the common factor of $3$ .

Cancel the common factor.

$512 x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Rewrite the expression.

$512 x^{1} = {(- 1)}^{3}$

Simplify.

$512 x = {(- 1)}^{3}$

Simplify the right side.

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

$512 x = - 1$

Divide each term in $512 x = - 1$ by $512$ and simplify.

Divide each term in $512 x = - 1$ by $512$ .

$\frac{512 x}{512} = \frac{- 1}{512}$

Simplify the left side.

Cancel the common factor of $512$ .

Cancel the common factor.

$\frac{512 x}{512} = \frac{- 1}{512}$

Divide $x$ by $1$ .

$x = \frac{- 1}{512}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{512}$

Step 14

The final solution is all the values that make $\frac{1 - 7 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} \cdot \frac{1 + 8 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} = 0$ true.

$x = \frac{1}{343}, - \frac{1}{512}$

Solve by Factoring x^(-2/3)+x^(-1/3)-56=0

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