Precalculus Examples

$x^{6} - 16^{4} = 0$

Step 1

Rewrite $x^{6}$ as ${(x^{3})}^{2}$ .

${(x^{3})}^{2} - 16^{4} = 0$

Step 2

Rewrite $16^{4}$ as ${(16^{2})}^{2}$ .

${(x^{3})}^{2} - {(16^{2})}^{2} = 0$

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x^{3}$ and $b = 16^{2}$ .

$(x^{3} + 16^{2}) (x^{3} - 16^{2}) = 0$

Step 4

Simplify.

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Step 4.1

Raise $16$ to the power of $2$ .

$(x^{3} + 256) (x^{3} - 16^{2}) = 0$

Step 4.2

Raise $16$ to the power of $2$ .

$(x^{3} + 256) (x^{3} - 1 \cdot 256) = 0$

Step 4.3

Multiply $- 1$ by $256$ .

$(x^{3} + 256) (x^{3} - 256) = 0$

Step 5

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

$x^{3} + 256 = 0$

$x^{3} - 256 = 0$

Step 6

Set $x^{3} + 256$ equal to $0$ and solve for $x$ .

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Step 6.1

Set $x^{3} + 256$ equal to $0$ .

$x^{3} + 256 = 0$

Step 6.2

Solve $x^{3} + 256 = 0$ for $x$ .

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Step 6.2.1

Subtract $256$ from both sides of the equation.

$x^{3} = - 256$

Step 6.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{- 256}$

Step 6.2.3

Simplify $\sqrt[3]{- 256}$ .

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Step 6.2.3.1

Rewrite $- 256$ as ${(- 4)}^{3} \cdot 4$ .

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Step 6.2.3.1.1

Factor $- 64$ out of $- 256$ .

$x = \sqrt[3]{- 64 (4)}$

Step 6.2.3.1.2

Rewrite $- 64$ as ${(- 4)}^{3}$ .

$x = \sqrt[3]{{(- 4)}^{3} \cdot 4}$

Step 6.2.3.2

Pull terms out from under the radical.

$x = - 4 \sqrt[3]{4}$

Step 7

Set $x^{3} - 256$ equal to $0$ and solve for $x$ .

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Step 7.1

Set $x^{3} - 256$ equal to $0$ .

$x^{3} - 256 = 0$

Step 7.2

Solve $x^{3} - 256 = 0$ for $x$ .

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Step 7.2.1

Add $256$ to both sides of the equation.

$x^{3} = 256$

Step 7.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{256}$

Step 7.2.3

Simplify $\sqrt[3]{256}$ .

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Step 7.2.3.1

Rewrite $256$ as $4^{3} \cdot 4$ .

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Step 7.2.3.1.1

Factor $64$ out of $256$ .

$x = \sqrt[3]{64 (4)}$

Step 7.2.3.1.2

Rewrite $64$ as $4^{3}$ .

$x = \sqrt[3]{4^{3} \cdot 4}$

Step 7.2.3.2

Pull terms out from under the radical.

$x = 4 \sqrt[3]{4}$

Step 8

The final solution is all the values that make $(x^{3} + 256) (x^{3} - 256) = 0$ true.

$x = - 4 \sqrt[3]{4}, 4 \sqrt[3]{4}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = - 4 \sqrt[3]{4}, 4 \sqrt[3]{4}$

Decimal Form:

$x = - 6.34960420 \dots, 6.34960420 \dots$