Algebra Examples

$x^{- \frac{2}{3}} + x^{- \frac{1}{3}} - 20 = 0$

Step 1

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

${(x^{- \frac{1}{3}})}^{2} + x^{- \frac{1}{3}} - 20 = 0$

Step 2

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

$u^{2} + u - 20 = 0$

Step 3

Factor $u^{2} + u - 20$ using the AC method.

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Step 3.1

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 $- 20$ and whose sum is $1$ .

$- 4, 5$

Step 3.2

Write the factored form using these integers.

$(u - 4) (u + 5) = 0$

Step 4

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

$(x^{- \frac{1}{3}} - 4) (x^{- \frac{1}{3}} + 5) = 0$

Step 5

Simplify.

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Step 5.1

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

$(\frac{1}{x^{\frac{1}{3}}} - 4) (x^{- \frac{1}{3}} + 5) = 0$

Step 5.2

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

$(\frac{1}{x^{\frac{1}{3}}} - 4) (\frac{1}{x^{\frac{1}{3}}} + 5) = 0$

Step 6

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ .

$(\frac{1}{x^{\frac{1}{3}}} - 4 \cdot \frac{x^{\frac{1}{3}}}{x^{\frac{1}{3}}}) (\frac{1}{x^{\frac{1}{3}}} + 5) = 0$

Step 7

Combine $- 4$ and $\frac{x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ .

$(\frac{1}{x^{\frac{1}{3}}} + \frac{- 4 x^{\frac{1}{3}}}{x^{\frac{1}{3}}}) (\frac{1}{x^{\frac{1}{3}}} + 5) = 0$

Step 8

Combine the numerators over the common denominator.

$\frac{1 - 4 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} \cdot (\frac{1}{x^{\frac{1}{3}}} + 5) = 0$

Step 9

To write $5$ as a fraction with a common denominator, multiply by $\frac{x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ .

$\frac{1 - 4 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} \cdot (\frac{1}{x^{\frac{1}{3}}} + \frac{5 x^{\frac{1}{3}}}{x^{\frac{1}{3}}}) = 0$

Step 10

Combine the numerators over the common denominator.

$\frac{1 - 4 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} \cdot \frac{1 + 5 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} = 0$

Step 11

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

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

$\frac{1 + 5 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} = 0$

Step 12

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

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Step 12.1

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

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

Step 12.2

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

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Step 12.2.1

Set the numerator equal to zero.

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

Step 12.2.2

Solve the equation for $x$ .

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Step 12.2.2.1

Subtract $1$ from both sides of the equation.

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

Step 12.2.2.2

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

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

Step 12.2.2.3

Simplify the exponent.

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Step 12.2.2.3.1

Simplify the left side.

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Step 12.2.2.3.1.1

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

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Step 12.2.2.3.1.1.1

Apply the product rule to $- 4 x^{\frac{1}{3}}$ .

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

Step 12.2.2.3.1.1.2

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

$- 64 {(x^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Step 12.2.2.3.1.1.3

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

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Step 12.2.2.3.1.1.3.1

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

$- 64 x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Step 12.2.2.3.1.1.3.2

Cancel the common factor of $3$ .

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Step 12.2.2.3.1.1.3.2.1

Cancel the common factor.

$- 64 x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Step 12.2.2.3.1.1.3.2.2

Rewrite the expression.

$- 64 x^{1} = {(- 1)}^{3}$

Step 12.2.2.3.1.1.4

Simplify.

$- 64 x = {(- 1)}^{3}$

Step 12.2.2.3.2

Simplify the right side.

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Step 12.2.2.3.2.1

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

$- 64 x = - 1$

Step 12.2.2.4

Divide each term in $- 64 x = - 1$ by $- 64$ and simplify.

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Step 12.2.2.4.1

Divide each term in $- 64 x = - 1$ by $- 64$ .

$\frac{- 64 x}{- 64} = \frac{- 1}{- 64}$

Step 12.2.2.4.2

Simplify the left side.

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Step 12.2.2.4.2.1

Cancel the common factor of $- 64$ .

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Step 12.2.2.4.2.1.1

Cancel the common factor.

$\frac{- 64 x}{- 64} = \frac{- 1}{- 64}$

Step 12.2.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{- 64}$

Step 12.2.2.4.3

Simplify the right side.

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Step 12.2.2.4.3.1

Dividing two negative values results in a positive value.

$x = \frac{1}{64}$

Step 13

Set $\frac{1 + 5 x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ equal to $0$ and solve for $x$ .

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Step 13.1

Set $\frac{1 + 5 x^{\frac{1}{3}}}{x^{\frac{1}{3}}}$ equal to $0$ .

$\frac{1 + 5 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} = 0$

Step 13.2

Solve $\frac{1 + 5 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} = 0$ for $x$ .

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Step 13.2.1

Set the numerator equal to zero.

$1 + 5 x^{\frac{1}{3}} = 0$

Step 13.2.2

Solve the equation for $x$ .

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Step 13.2.2.1

Subtract $1$ from both sides of the equation.

$5 x^{\frac{1}{3}} = - 1$

Step 13.2.2.2

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

${(5 x^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Step 13.2.2.3

Simplify the exponent.

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Step 13.2.2.3.1

Simplify the left side.

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Step 13.2.2.3.1.1

Simplify ${(5 x^{\frac{1}{3}})}^{3}$ .

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Step 13.2.2.3.1.1.1

Apply the product rule to $5 x^{\frac{1}{3}}$ .

$5^{3} {(x^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Step 13.2.2.3.1.1.2

Raise $5$ to the power of $3$ .

$125 {(x^{\frac{1}{3}})}^{3} = {(- 1)}^{3}$

Step 13.2.2.3.1.1.3

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

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Step 13.2.2.3.1.1.3.1

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

$125 x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Step 13.2.2.3.1.1.3.2

Cancel the common factor of $3$ .

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Step 13.2.2.3.1.1.3.2.1

Cancel the common factor.

$125 x^{\frac{1}{3} \cdot 3} = {(- 1)}^{3}$

Step 13.2.2.3.1.1.3.2.2

Rewrite the expression.

$125 x^{1} = {(- 1)}^{3}$

Step 13.2.2.3.1.1.4

Simplify.

$125 x = {(- 1)}^{3}$

Step 13.2.2.3.2

Simplify the right side.

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Step 13.2.2.3.2.1

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

$125 x = - 1$

Step 13.2.2.4

Divide each term in $125 x = - 1$ by $125$ and simplify.

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Step 13.2.2.4.1

Divide each term in $125 x = - 1$ by $125$ .

$\frac{125 x}{125} = \frac{- 1}{125}$

Step 13.2.2.4.2

Simplify the left side.

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Step 13.2.2.4.2.1

Cancel the common factor of $125$ .

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Step 13.2.2.4.2.1.1

Cancel the common factor.

$\frac{125 x}{125} = \frac{- 1}{125}$

Step 13.2.2.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{125}$

Step 13.2.2.4.3

Simplify the right side.

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Step 13.2.2.4.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{125}$

Step 14

The final solution is all the values that make $\frac{1 - 4 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} \cdot \frac{1 + 5 x^{\frac{1}{3}}}{x^{\frac{1}{3}}} = 0$ true.

$x = \frac{1}{64}, - \frac{1}{125}$