Precalculus Examples

$x^{\frac{4}{3}} + 3 x^{\frac{2}{3}} - 28 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Rewrite $x^{\frac{4}{3}}$ as ${(x^{\frac{2}{3}})}^{2}$ .

${(x^{\frac{2}{3}})}^{2} + 3 x^{\frac{2}{3}} - 28 = 0$

Step 1.2

Let $u = x^{\frac{2}{3}}$ . Substitute $u$ for all occurrences of $x^{\frac{2}{3}}$ .

$u^{2} + 3 u - 28 = 0$

Step 1.3

Factor $u^{2} + 3 u - 28$ 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 $- 28$ and whose sum is $3$ .

$- 4, 7$

Step 1.3.2

Write the factored form using these integers.

$(u - 4) (u + 7) = 0$

Step 1.4

Replace all occurrences of $u$ with $x^{\frac{2}{3}}$ .

$(x^{\frac{2}{3}} - 4) (x^{\frac{2}{3}} + 7) = 0$

Step 2

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

$x^{\frac{2}{3}} - 4 = 0$

$x^{\frac{2}{3}} + 7 = 0$

Step 3

Set $x^{\frac{2}{3}} - 4$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x^{\frac{2}{3}} - 4$ equal to $0$ .

$x^{\frac{2}{3}} - 4 = 0$

Step 3.2

Solve $x^{\frac{2}{3}} - 4 = 0$ for $x$ .

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Step 3.2.1

Add $4$ to both sides of the equation.

$x^{\frac{2}{3}} = 4$

Step 3.2.2

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{2}{3}})}^{\frac{3}{2}} = \pm 4^{\frac{3}{2}}$

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

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Step 3.2.3.1.1.1

Multiply the exponents in ${(x^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 3.2.3.1.1.1.1

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

$x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 4^{\frac{3}{2}}$

Step 3.2.3.1.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.3.1.1.1.2.1

Cancel the common factor.

$x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm 4^{\frac{3}{2}}$

Step 3.2.3.1.1.1.2.2

Rewrite the expression.

$x^{\frac{1}{3} \cdot 3} = \pm 4^{\frac{3}{2}}$

Step 3.2.3.1.1.1.3

Cancel the common factor of $3$ .

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Step 3.2.3.1.1.1.3.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = \pm 4^{\frac{3}{2}}$

Step 3.2.3.1.1.1.3.2

Rewrite the expression.

$x^{1} = \pm 4^{\frac{3}{2}}$

Step 3.2.3.1.1.2

Simplify.

$x = \pm 4^{\frac{3}{2}}$

Step 3.2.3.2

Simplify the right side.

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Step 3.2.3.2.1

Simplify $\pm 4^{\frac{3}{2}}$ .

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Step 3.2.3.2.1.1

Simplify the expression.

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Step 3.2.3.2.1.1.1

Rewrite $4$ as $2^{2}$ .

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

Step 3.2.3.2.1.1.2

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

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

Step 3.2.3.2.1.2

Cancel the common factor of $2$ .

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Step 3.2.3.2.1.2.1

Cancel the common factor.

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

Step 3.2.3.2.1.2.2

Rewrite the expression.

$x = \pm 2^{3}$

Step 3.2.3.2.1.3

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

$x = \pm 8$

Step 3.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 3.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 8$

Step 3.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 8$

Step 3.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 8, - 8$

Step 4

Set $x^{\frac{2}{3}} + 7$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x^{\frac{2}{3}} + 7$ equal to $0$ .

$x^{\frac{2}{3}} + 7 = 0$

Step 4.2

Solve $x^{\frac{2}{3}} + 7 = 0$ for $x$ .

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Step 4.2.1

Subtract $7$ from both sides of the equation.

$x^{\frac{2}{3}} = - 7$

Step 4.2.2

Raise each side of the equation to the power of $\frac{3}{2}$ to eliminate the fractional exponent on the left side.

${(x^{\frac{2}{3}})}^{\frac{3}{2}} = \pm {(- 7)}^{\frac{3}{2}}$

Step 4.2.3

Simplify the left side.

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Step 4.2.3.1

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

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Step 4.2.3.1.1

Multiply the exponents in ${(x^{\frac{2}{3}})}^{\frac{3}{2}}$ .

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Step 4.2.3.1.1.1

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

$x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm {(- 7)}^{\frac{3}{2}}$

Step 4.2.3.1.1.2

Cancel the common factor of $2$ .

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Step 4.2.3.1.1.2.1

Cancel the common factor.

$x^{\frac{2}{3} \cdot \frac{3}{2}} = \pm {(- 7)}^{\frac{3}{2}}$

Step 4.2.3.1.1.2.2

Rewrite the expression.

$x^{\frac{1}{3} \cdot 3} = \pm {(- 7)}^{\frac{3}{2}}$

Step 4.2.3.1.1.3

Cancel the common factor of $3$ .

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Step 4.2.3.1.1.3.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = \pm {(- 7)}^{\frac{3}{2}}$

Step 4.2.3.1.1.3.2

Rewrite the expression.

$x^{1} = \pm {(- 7)}^{\frac{3}{2}}$

Step 4.2.3.1.2

Simplify.

$x = \pm {(- 7)}^{\frac{3}{2}}$

Step 4.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = {(- 7)}^{\frac{3}{2}}$

Step 4.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - {(- 7)}^{\frac{3}{2}}$

Step 4.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = {(- 7)}^{\frac{3}{2}}, - {(- 7)}^{\frac{3}{2}}$

Step 5

The final solution is all the values that make $(x^{\frac{2}{3}} - 4) (x^{\frac{2}{3}} + 7) = 0$ true.

$x = 8, - 8, {(- 7)}^{\frac{3}{2}}, - {(- 7)}^{\frac{3}{2}}$

Step 6

Exclude the solutions that do not make $x^{\frac{4}{3}} + 3 x^{\frac{2}{3}} - 28 = 0$ true.

$x = 8, - 8$