Basic Math Examples

$\frac{1}{a} + \frac{a}{1} + \frac{a}{a} + \frac{1}{1} = 11 b$

Step 1

Rewrite the equation as $11 b = \frac{1}{a} + \frac{a}{1} + \frac{a}{a} + \frac{1}{1}$ .

$11 b = \frac{1}{a} + \frac{a}{1} + \frac{a}{a} + \frac{1}{1}$

Step 2

Simplify.

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Step 2.1

Reduce the expression $\frac{a}{a}$ by cancelling the common factors.

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Step 2.1.1

Cancel the common factor.

$11 b = \frac{1}{a} + \frac{a}{1} + \frac{a}{a} + \frac{1}{1}$

Step 2.1.2

Rewrite the expression.

$11 b = \frac{1}{a} + \frac{a}{1} + \frac{1}{1} + \frac{1}{1}$

Step 2.2

Divide $a$ by $1$ .

$11 b = \frac{1}{a} + a + \frac{1}{1} + \frac{1}{1}$

Step 2.3

Rewrite the expression.

$11 b = \frac{1}{a} + a + 1 + \frac{1}{1}$

Step 2.4

Rewrite the expression.

$11 b = \frac{1}{a} + a + 1 + 1$

Step 3

Divide each term in $11 b = \frac{1}{a} + a + 1 + 1$ by $11$ and simplify.

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Step 3.1

Divide each term in $11 b = \frac{1}{a} + a + 1 + 1$ by $11$ .

$\frac{11 b}{11} = \frac{\frac{1}{a}}{11} + \frac{a}{11} + \frac{1}{11} + \frac{1}{11}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $11$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{11 b}{11} = \frac{\frac{1}{a}}{11} + \frac{a}{11} + \frac{1}{11} + \frac{1}{11}$

Step 3.2.1.2

Divide $b$ by $1$ .

$b = \frac{\frac{1}{a}}{11} + \frac{a}{11} + \frac{1}{11} + \frac{1}{11}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Combine fractions.

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Step 3.3.1.1

Combine the numerators over the common denominator.

$b = \frac{\frac{1}{a} + a + 1 + 1}{11}$

Step 3.3.1.2

Add $1$ and $1$ .

$b = \frac{\frac{1}{a} + a + 2}{11}$

Step 3.3.2

Simplify the numerator.

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Step 3.3.2.1

Reorder terms.

$b = \frac{a + \frac{1}{a} + 2}{11}$

Step 3.3.2.2

To write $a$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

$b = \frac{\frac{a \cdot a}{a} + \frac{1}{a} + 2}{11}$

Step 3.3.2.3

Combine the numerators over the common denominator.

$b = \frac{\frac{a \cdot a + 1}{a} + 2}{11}$

Step 3.3.2.4

Multiply $a$ by $a$ .

$b = \frac{\frac{a^{2} + 1}{a} + 2}{11}$

Step 3.3.2.5

To write $2$ as a fraction with a common denominator, multiply by $\frac{a}{a}$ .

$b = \frac{\frac{a^{2} + 1}{a} + \frac{2 a}{a}}{11}$

Step 3.3.2.6

Combine the numerators over the common denominator.

$b = \frac{\frac{a^{2} + 1 + 2 a}{a}}{11}$

Step 3.3.2.7

Factor using the perfect square rule.

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Step 3.3.2.7.1

Rearrange terms.

$b = \frac{\frac{a^{2} + 2 a + 1}{a}}{11}$

Step 3.3.2.7.2

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

$b = \frac{\frac{a^{2} + 2 a + 1^{2}}{a}}{11}$

Step 3.3.2.7.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 a = 2 \cdot a \cdot 1$

Step 3.3.2.7.4

Rewrite the polynomial.

$b = \frac{\frac{a^{2} + 2 \cdot a \cdot 1 + 1^{2}}{a}}{11}$

Step 3.3.2.7.5

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = a$ and $b = 1$ .

$b = \frac{\frac{{(a + 1)}^{2}}{a}}{11}$

Step 3.3.3

Multiply the numerator by the reciprocal of the denominator.

$b = \frac{{(a + 1)}^{2}}{a} \cdot \frac{1}{11}$

Step 3.3.4

Combine.

$b = \frac{{(a + 1)}^{2} \cdot 1}{a \cdot 11}$

Step 3.3.5

Simplify the expression.

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Step 3.3.5.1

Multiply ${(a + 1)}^{2}$ by $1$ .

$b = \frac{{(a + 1)}^{2}}{a \cdot 11}$

Step 3.3.5.2

Move $11$ to the left of $a$ .

$b = \frac{{(a + 1)}^{2}}{11 a}$