Algebra Examples

$p (x) = (\frac{1}{5} x) (- 4 x) (2 x)$

Step 1

Set $(\frac{1}{5} x) (- 4 x) (2 x)$ equal to $0$ .

$(\frac{1}{5} x) (- 4 x) (2 x) = 0$

Step 2

Solve for $x$ .

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Step 2.1

Multiply both sides of the equation by $\frac{1}{\frac{1}{5} \cdot - 4 \cdot 2}$ .

$\frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} (\frac{1}{5} \cdot (x (- 4 x) (2 x))) = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2

Simplify both sides of the equation.

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Step 2.2.1

Simplify the left side.

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Step 2.2.1.1

Simplify $\frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} (\frac{1}{5} \cdot (x (- 4 x) (2 x)))$ .

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Step 2.2.1.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 2.2.1.1.1.1

Move $x$ .

$\frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} (\frac{1}{5} \cdot (x \cdot x \cdot - 4 (2 x))) = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.1.2

Multiply $x$ by $x$ .

$\frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} (\frac{1}{5} \cdot (x^{2} \cdot - 4 (2 x))) = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.2

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 2.2.1.1.2.1

Move $x$ .

$\frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} (\frac{1}{5} \cdot (x \cdot x^{2} \cdot - 4 \cdot 2)) = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.2.2

Multiply $x$ by $x^{2}$ .

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Step 2.2.1.1.2.2.1

Raise $x$ to the power of $1$ .

$\frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} (\frac{1}{5} \cdot (x^{1} x^{2} \cdot - 4 \cdot 2)) = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} (\frac{1}{5} \cdot (x^{1 + 2} \cdot - 4 \cdot 2)) = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.2.3

Add $1$ and $2$ .

$\frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} (\frac{1}{5} \cdot (x^{3} \cdot - 4 \cdot 2)) = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.3

Simplify the denominator.

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Step 2.2.1.1.3.1

Combine $\frac{1}{5}$ and $- 4$ .

$\frac{1}{\frac{- 4}{5} \cdot 2} (\frac{1}{5} \cdot (x^{3} \cdot - 4 \cdot 2)) = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.3.2

Combine $\frac{- 4}{5}$ and $2$ .

$\frac{1}{\frac{- 4 \cdot 2}{5}} (\frac{1}{5} \cdot (x^{3} \cdot - 4 \cdot 2)) = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.4

Reduce the expression by cancelling the common factors.

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Step 2.2.1.1.4.1

Multiply $- 4$ by $2$ .

$\frac{1}{\frac{- 8}{5}} (\frac{1}{5} \cdot (x^{3} \cdot - 4 \cdot 2)) = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.4.2

Move the negative in front of the fraction.

$\frac{1}{- \frac{8}{5}} (\frac{1}{5} \cdot (x^{3} \cdot - 4 \cdot 2)) = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.4.3

Cancel the common factor of $1$ and $- 1$ .

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Step 2.2.1.1.4.3.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- 1 (- 1)}{- \frac{8}{5}} (\frac{1}{5} \cdot (x^{3} \cdot - 4 \cdot 2)) = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.4.3.2

Move the negative in front of the fraction.

$- \frac{1}{\frac{8}{5}} (\frac{1}{5} \cdot (x^{3} \cdot - 4 \cdot 2)) = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.5

Multiply the numerator by the reciprocal of the denominator.

$- (1 (\frac{5}{8})) (\frac{1}{5} \cdot (x^{3} \cdot - 4 \cdot 2)) = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.6

Simplify the expression.

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Step 2.2.1.1.6.1

Multiply $\frac{5}{8}$ by $1$ .

$- \frac{5}{8} (\frac{1}{5} \cdot (x^{3} \cdot - 4 \cdot 2)) = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.6.2

Move $- 4$ to the left of $x^{3}$ .

$- \frac{5}{8} (\frac{1}{5} \cdot (- 4 \cdot x^{3} \cdot 2)) = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.6.3

Multiply $2$ by $- 4$ .

$- \frac{5}{8} (\frac{1}{5} \cdot (- 8 x^{3})) = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.7

Cancel the common factor of $8$ .

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Step 2.2.1.1.7.1

Move the leading negative in $- \frac{5}{8}$ into the numerator.

$\frac{- 5}{8} (\frac{1}{5} \cdot (- 8 x^{3})) = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.7.2

Factor $8$ out of $\frac{1}{5} \cdot (- 8 x^{3})$ .

$\frac{- 5}{8} (8 (\frac{1}{5} \cdot (- x^{3}))) = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.7.3

Cancel the common factor.

$\frac{- 5}{8} (8 (\frac{1}{5} \cdot (- x^{3}))) = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.7.4

Rewrite the expression.

$- 5 (\frac{1}{5} \cdot (- x^{3})) = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.8

Combine $\frac{1}{5}$ and $x^{3}$ .

$- 5 (- \frac{x^{3}}{5}) = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.9

Multiply $- 1$ by $- 5$ .

$5 \frac{x^{3}}{5} = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.10

Combine $5$ and $\frac{x^{3}}{5}$ .

$\frac{5 x^{3}}{5} = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.11

Cancel the common factor of $5$ .

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Step 2.2.1.1.11.1

Cancel the common factor.

$\frac{5 x^{3}}{5} = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.1.1.11.2

Divide $x^{3}$ by $1$ .

$x^{3} = \frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$

Step 2.2.2

Simplify the right side.

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Step 2.2.2.1

Simplify $\frac{1}{\frac{1}{5} \cdot - 4 \cdot 2} \cdot 0$ .

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Step 2.2.2.1.1

Simplify the denominator.

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Step 2.2.2.1.1.1

Combine $\frac{1}{5}$ and $- 4$ .

$x^{3} = \frac{1}{\frac{- 4}{5} \cdot 2} \cdot 0$

Step 2.2.2.1.1.2

Combine $\frac{- 4}{5}$ and $2$ .

$x^{3} = \frac{1}{\frac{- 4 \cdot 2}{5}} \cdot 0$

Step 2.2.2.1.2

Reduce the expression by cancelling the common factors.

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Step 2.2.2.1.2.1

Multiply $- 4$ by $2$ .

$x^{3} = \frac{1}{\frac{- 8}{5}} \cdot 0$

Step 2.2.2.1.2.2

Move the negative in front of the fraction.

$x^{3} = \frac{1}{- \frac{8}{5}} \cdot 0$

Step 2.2.2.1.2.3

Cancel the common factor of $1$ and $- 1$ .

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Step 2.2.2.1.2.3.1

Rewrite $1$ as $- 1 (- 1)$ .

$x^{3} = \frac{- 1 (- 1)}{- \frac{8}{5}} \cdot 0$

Step 2.2.2.1.2.3.2

Move the negative in front of the fraction.

$x^{3} = - \frac{1}{\frac{8}{5}} \cdot 0$

Step 2.2.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$x^{3} = - (1 (\frac{5}{8})) \cdot 0$

Step 2.2.2.1.4

Multiply $\frac{5}{8}$ by $1$ .

$x^{3} = - \frac{5}{8} \cdot 0$

Step 2.2.2.1.5

Multiply $- \frac{5}{8} \cdot 0$ .

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Step 2.2.2.1.5.1

Multiply $0$ by $- 1$ .

$x^{3} = 0 (\frac{5}{8})$

Step 2.2.2.1.5.2

Multiply $0$ by $\frac{5}{8}$ .

$x^{3} = 0$

Step 2.3

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

$x = \sqrt[3]{0}$

Step 2.4

Simplify $\sqrt[3]{0}$ .

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Step 2.4.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 2.4.2

Pull terms out from under the radical, assuming real numbers.

$x = 0$

Step 3