Precalculus Examples

x^{\frac{4}{3}} - 5 x^{\frac{2}{3}} + 6 = 0

$x^{\frac{4}{3}} - 5 x^{\frac{2}{3}} + 6 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Rewrite

x^{\frac{4}{3}}

$x^{\frac{4}{3}}$ as

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

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

{(x^{\frac{2}{3}})}^{2} - 5 x^{\frac{2}{3}} + 6 = 0

${(x^{\frac{2}{3}})}^{2} - 5 x^{\frac{2}{3}} + 6 = 0$

Step 1.2

Let

u = x^{\frac{2}{3}}

$u = x^{\frac{2}{3}}$ . Substitute

u

$u$ for all occurrences of

x^{\frac{2}{3}}

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

u^{2} - 5 u + 6 = 0

$u^{2} - 5 u + 6 = 0$

Step 1.3

Factor

u^{2} - 5 u + 6

$u^{2} - 5 u + 6$ using the AC method.

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Step 1.3.1

Consider the form

x^{2} + b x + c

$x^{2} + b x + c$ . Find a pair of integers whose product is

c

$c$ and whose sum is

b

$b$ . In this case, whose product is

6

$6$ and whose sum is

- 5

$- 5$ .

- 3, - 2

$- 3, - 2$

Step 1.3.2

Write the factored form using these integers.

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

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

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

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

Step 1.4

Replace all occurrences of

u

$u$ with

x^{\frac{2}{3}}

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

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

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

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

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

Step 2

If any individual factor on the left side of the equation is equal to

0

$0$ , the entire expression will be equal to

0

$0$ .

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

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

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

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

Step 3

Set

x^{\frac{2}{3}} - 3

$x^{\frac{2}{3}} - 3$ equal to

0

$0$ and solve for

x

$x$ .

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Step 3.1

Set

x^{\frac{2}{3}} - 3

$x^{\frac{2}{3}} - 3$ equal to

0

$0$ .

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

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

Step 3.2

Solve

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

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

x

$x$ .

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Step 3.2.1

Add

3

$3$ to both sides of the equation.

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

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

Step 3.2.2

Raise each side of the equation to the power of

\frac{3}{2}

$\frac{3}{2}$ to eliminate the fractional exponent on the left side.

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

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

Step 3.2.3

Simplify the left side.

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Step 3.2.3.1

Simplify

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

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

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Step 3.2.3.1.1

Multiply the exponents in

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

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

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Step 3.2.3.1.1.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

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

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

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

Step 3.2.3.1.1.2

Cancel the common factor of

2

$2$ .

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Step 3.2.3.1.1.2.1

Cancel the common factor.

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

Step 3.2.3.1.1.2.2

Rewrite the expression.

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

Step 3.2.3.1.1.3

Cancel the common factor of

$3$ .

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Step 3.2.3.1.1.3.1

Cancel the common factor.

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

Step 3.2.3.1.1.3.2

Rewrite the expression.

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

Step 3.2.3.1.2

Simplify.

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

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.

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

Step 3.2.4.2

Next, use the negative value of the

$\pm$ to find the second solution.

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

Step 3.2.4.3

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

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

Step 4

Set

$x^{\frac{2}{3}} - 2$ equal to

$0$ and solve for

$x$ .

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Step 4.1

Set

$x^{\frac{2}{3}} - 2$ equal to

$0$ .

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

Step 4.2

Solve

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

$x$ .

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Step 4.2.1

Add

$2$ to both sides of the equation.

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

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.

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

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,

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

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

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.

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

Step 4.2.3.1.1.2.2

Rewrite the expression.

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

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.

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

Step 4.2.3.1.1.3.2

Rewrite the expression.

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

Step 4.2.3.1.2

Simplify.

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

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.

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

Step 4.2.4.2

Next, use the negative value of the

$\pm$ to find the second solution.

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

Step 4.2.4.3

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

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

Step 5

The final solution is all the values that make

$(x^{\frac{2}{3}} - 3) (x^{\frac{2}{3}} - 2) = 0$ true.

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

Step 6

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 5.19615242 \dots, - 5.19615242 \dots, 2.82842712 \dots, - 2.82842712 \dots$