Algebra Examples

$x^{- \frac{2}{3}} + x^{- \frac{1}{3}} - 6 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

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

${(x^{- \frac{1}{3}})}^{2} + x^{- \frac{1}{3}} - 6 = 0$

Step 1.2

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

$u^{2} + u - 6 = 0$

Step 1.3

Factor $u^{2} + u - 6$ using the AC method.

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Step 1.3.1

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 $- 6$ and whose sum is $1$ .

$- 2, 3$

Step 1.3.2

Write the factored form using these integers.

$(u - 2) (u + 3) = 0$

Step 1.4

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

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

Step 2

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

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

Add $2$ to both sides of the equation.

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

Step 3.2.2

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

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

Step 3.2.3

Simplify the exponent.

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Step 3.2.3.1

Simplify the left side.

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Step 3.2.3.1.1

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

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Step 3.2.3.1.1.1

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

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Step 3.2.3.1.1.1.1

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

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

Step 3.2.3.1.1.1.2

Cancel the common factor of $3$ .

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Step 3.2.3.1.1.1.2.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

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

Step 3.2.3.1.1.1.2.2

Factor $3$ out of $- 3$ .

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

Step 3.2.3.1.1.1.2.3

Cancel the common factor.

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

Step 3.2.3.1.1.1.2.4

Rewrite the expression.

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

Step 3.2.3.1.1.1.3

Multiply $- 1$ by $- 1$ .

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

Step 3.2.3.1.1.2

Simplify.

$x = 2^{- 3}$

Step 3.2.3.2

Simplify the right side.

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Step 3.2.3.2.1

Simplify $2^{- 3}$ .

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Step 3.2.3.2.1.1

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

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

Step 3.2.3.2.1.2

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

Subtract $3$ from both sides of the equation.

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

Step 4.2.2

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

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

Step 4.2.3

Simplify the exponent.

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Step 4.2.3.1

Simplify the left side.

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Step 4.2.3.1.1

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

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Step 4.2.3.1.1.1

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

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Step 4.2.3.1.1.1.1

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

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

Step 4.2.3.1.1.1.2

Cancel the common factor of $3$ .

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Step 4.2.3.1.1.1.2.1

Move the leading negative in $- \frac{1}{3}$ into the numerator.

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

Step 4.2.3.1.1.1.2.2

Factor $3$ out of $- 3$ .

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

Step 4.2.3.1.1.1.2.3

Cancel the common factor.

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

Step 4.2.3.1.1.1.2.4

Rewrite the expression.

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

Step 4.2.3.1.1.1.3

Multiply $- 1$ by $- 1$ .

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

Step 4.2.3.1.1.2

Simplify.

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

Step 4.2.3.2

Simplify the right side.

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Step 4.2.3.2.1

Simplify ${(- 3)}^{- 3}$ .

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Step 4.2.3.2.1.1

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

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

Step 4.2.3.2.1.2

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

$x = \frac{1}{- 27}$

Step 4.2.3.2.1.3

Move the negative in front of the fraction.

$x = - \frac{1}{27}$

Step 5

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

$x = \frac{1}{8}, - \frac{1}{27}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{8}, - \frac{1}{27}$

Decimal Form:

$x = 0.125, - 0.\overline{037}$