Precalculus Examples

${(x)}^{- \frac{2}{3}} + {(x)}^{- \frac{1}{3}} - 12 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

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

${(x^{- \frac{1}{3}})}^{2} + x^{- \frac{1}{3}} - 12 = 0$

Step 1.2

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

$u^{2} + u - 12 = 0$

Step 1.3

Factor $u^{2} + u - 12$ 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 $- 12$ and whose sum is $1$ .

$- 3, 4$

Step 1.3.2

Write the factored form using these integers.

$(u - 3) (u + 4) = 0$

Step 1.4

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

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

Step 2

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

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

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

Step 3

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

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

Add $3$ to both sides of the equation.

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

Step 3.2.2

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

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

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

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

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.

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

Step 3.2.3.1.1.1.2.2

Factor $3$ out of $- 3$ .

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

Step 3.2.3.1.1.1.2.3

Cancel the common factor.

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

Step 3.2.3.1.1.1.2.4

Rewrite the expression.

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

Step 3.2.3.1.1.1.3

Multiply $- 1$ by $- 1$ .

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

Step 3.2.3.1.1.2

Simplify.

$x = 3^{- 3}$

Step 3.2.3.2

Simplify the right side.

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Step 3.2.3.2.1

Simplify $3^{- 3}$ .

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Step 3.2.3.2.1.1

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

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

Step 3.2.3.2.1.2

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

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

Step 4

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

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Step 4.1

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

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

Step 4.2

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

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Step 4.2.1

Subtract $4$ from both sides of the equation.

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

Step 4.2.2

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

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

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

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

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.

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

Step 4.2.3.1.1.1.2.2

Factor $3$ out of $- 3$ .

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

Step 4.2.3.1.1.1.2.3

Cancel the common factor.

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

Step 4.2.3.1.1.1.2.4

Rewrite the expression.

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

Step 4.2.3.1.1.1.3

Multiply $- 1$ by $- 1$ .

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

Step 4.2.3.1.1.2

Simplify.

$x = {(- 4)}^{- 3}$

Step 4.2.3.2

Simplify the right side.

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Step 4.2.3.2.1

Simplify ${(- 4)}^{- 3}$ .

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Step 4.2.3.2.1.1

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

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

Step 4.2.3.2.1.2

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

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

Step 4.2.3.2.1.3

Move the negative in front of the fraction.

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

Step 5

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

$x = \frac{1}{27}, - \frac{1}{64}$

Step 6