Pre-Algebra Examples

$- \frac{\frac{a + x}{b}}{\frac{1}{a}} = a b$

Step 1

Divide each term in $- \frac{\frac{a + x}{b}}{\frac{1}{a}} = a b$ by $- 1$ and simplify.

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Step 1.1

Divide each term in $- \frac{\frac{a + x}{b}}{\frac{1}{a}} = a b$ by $- 1$ .

$\frac{- \frac{\frac{a + x}{b}}{\frac{1}{a}}}{- 1} = \frac{a b}{- 1}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Dividing two negative values results in a positive value.

$\frac{\frac{\frac{a + x}{b}}{\frac{1}{a}}}{1} = \frac{a b}{- 1}$

Step 1.2.2

Divide $\frac{\frac{a + x}{b}}{\frac{1}{a}}$ by $1$ .

$\frac{\frac{a + x}{b}}{\frac{1}{a}} = \frac{a b}{- 1}$

Step 1.2.3

Multiply the numerator by the reciprocal of the denominator.

$\frac{a + x}{b} a = \frac{a b}{- 1}$

Step 1.2.4

Combine $\frac{a + x}{b}$ and $a$ .

$\frac{(a + x) a}{b} = \frac{a b}{- 1}$

Step 1.2.5

Reorder factors in $\frac{(a + x) a}{b}$ .

$\frac{a (a + x)}{b} = \frac{a b}{- 1}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Move the negative one from the denominator of $\frac{a b}{- 1}$ .

$\frac{a (a + x)}{b} = - 1 \cdot (a b)$

Step 1.3.2

Rewrite $- 1 \cdot (a b)$ as $- (a b)$ .

$\frac{a (a + x)}{b} = - a b$

Step 2

Multiply both sides by $b$ .

$\frac{a (a + x)}{b} b = - a b \cdot b$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{a (a + x)}{b} b$ .

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Step 3.1.1.1

Cancel the common factor of $b$ .

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Step 3.1.1.1.1

Cancel the common factor.

$\frac{a (a + x)}{b} b = - a b \cdot b$

Step 3.1.1.1.2

Rewrite the expression.

$a (a + x) = - a b \cdot b$

Step 3.1.1.2

Apply the distributive property.

$a \cdot a + a x = - a b \cdot b$

Step 3.1.1.3

Simplify the expression.

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Step 3.1.1.3.1

Multiply $a$ by $a$ .

$a^{2} + a x = - a b \cdot b$

Step 3.1.1.3.2

Reorder $a^{2}$ and $a x$ .

$a x + a^{2} = - a b \cdot b$

Step 3.2

Simplify the right side.

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Step 3.2.1

Multiply $b$ by $b$ by adding the exponents.

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Step 3.2.1.1

Move $b$ .

$a x + a^{2} = - a (b \cdot b)$

Step 3.2.1.2

Multiply $b$ by $b$ .

$a x + a^{2} = - a b^{2}$

Step 4

Solve for $x$ .

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Step 4.1

Subtract $a^{2}$ from both sides of the equation.

$a x = - a b^{2} - a^{2}$

Step 4.2

Divide each term in $a x = - a b^{2} - a^{2}$ by $a$ and simplify.

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Step 4.2.1

Divide each term in $a x = - a b^{2} - a^{2}$ by $a$ .

$\frac{a x}{a} = \frac{- a b^{2}}{a} + \frac{- a^{2}}{a}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $a$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{a x}{a} = \frac{- a b^{2}}{a} + \frac{- a^{2}}{a}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- a b^{2}}{a} + \frac{- a^{2}}{a}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Simplify each term.

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Step 4.2.3.1.1

Cancel the common factor of $a$ .

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Step 4.2.3.1.1.1

Cancel the common factor.

$x = \frac{- a b^{2}}{a} + \frac{- a^{2}}{a}$

Step 4.2.3.1.1.2

Divide $- 1 b^{2}$ by $1$ .

$x = - 1 b^{2} + \frac{- a^{2}}{a}$

Step 4.2.3.1.2

Rewrite $- 1 b^{2}$ as $- b^{2}$ .

$x = - b^{2} + \frac{- a^{2}}{a}$

Step 4.2.3.1.3

Cancel the common factor of $a^{2}$ and $a$ .

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Step 4.2.3.1.3.1

Factor $a$ out of $- a^{2}$ .

$x = - b^{2} + \frac{a (- a)}{a}$

Step 4.2.3.1.3.2

Cancel the common factors.

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Step 4.2.3.1.3.2.1

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

$x = - b^{2} + \frac{a (- a)}{a^{1}}$

Step 4.2.3.1.3.2.2

Factor $a$ out of $a^{1}$ .

$x = - b^{2} + \frac{a (- a)}{a \cdot 1}$

Step 4.2.3.1.3.2.3

Cancel the common factor.

$x = - b^{2} + \frac{a (- a)}{a \cdot 1}$

Step 4.2.3.1.3.2.4

Rewrite the expression.

$x = - b^{2} + \frac{- a}{1}$

Step 4.2.3.1.3.2.5

Divide $- a$ by $1$ .

$x = - b^{2} - a$