Basic Math Examples

$\frac{5}{2 {(y - 1)}^{2}} - \frac{3}{2 y^{2} - 2}$

Step 1

Simplify the denominator.

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Step 1.1

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

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Step 1.1.1

Factor $2$ out of $2 y^{2}$ .

$\frac{5}{2 {(y - 1)}^{2}} - \frac{3}{2 (y^{2}) - 2}$

Step 1.1.2

Factor $2$ out of $- 2$ .

$\frac{5}{2 {(y - 1)}^{2}} - \frac{3}{2 (y^{2}) + 2 (- 1)}$

Step 1.1.3

Factor $2$ out of $2 (y^{2}) + 2 (- 1)$ .

$\frac{5}{2 {(y - 1)}^{2}} - \frac{3}{2 (y^{2} - 1)}$

Step 1.2

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

$\frac{5}{2 {(y - 1)}^{2}} - \frac{3}{2 (y^{2} - 1^{2})}$

Step 1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = y$ and $b = 1$ .

$\frac{5}{2 {(y - 1)}^{2}} - \frac{3}{2 (y + 1) (y - 1)}$

Step 2

To write $\frac{5}{2 {(y - 1)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{y + 1}{y + 1}$ .

$\frac{5}{2 {(y - 1)}^{2}} \cdot \frac{y + 1}{y + 1} - \frac{3}{2 (y + 1) (y - 1)}$

Step 3

To write $- \frac{3}{2 (y + 1) (y - 1)}$ as a fraction with a common denominator, multiply by $\frac{y - 1}{y - 1}$ .

$\frac{5}{2 {(y - 1)}^{2}} \cdot \frac{y + 1}{y + 1} - \frac{3}{2 (y + 1) (y - 1)} \cdot \frac{y - 1}{y - 1}$

Step 4

Write each expression with a common denominator of $2 (y + 1) {(y - 1)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{5}{2 {(y - 1)}^{2}}$ by $\frac{y + 1}{y + 1}$ .

$\frac{5 (y + 1)}{2 {(y - 1)}^{2} (y + 1)} - \frac{3}{2 (y + 1) (y - 1)} \cdot \frac{y - 1}{y - 1}$

Step 4.2

Multiply $\frac{3}{2 (y + 1) (y - 1)}$ by $\frac{y - 1}{y - 1}$ .

$\frac{5 (y + 1)}{2 {(y - 1)}^{2} (y + 1)} - \frac{3 (y - 1)}{2 (y + 1) (y - 1) (y - 1)}$

Step 4.3

Raise $y - 1$ to the power of $1$ .

$\frac{5 (y + 1)}{2 {(y - 1)}^{2} (y + 1)} - \frac{3 (y - 1)}{2 (y + 1) ({(y - 1)}^{1} (y - 1))}$

Step 4.4

Raise $y - 1$ to the power of $1$ .

$\frac{5 (y + 1)}{2 {(y - 1)}^{2} (y + 1)} - \frac{3 (y - 1)}{2 (y + 1) ({(y - 1)}^{1} {(y - 1)}^{1})}$

Step 4.5

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

$\frac{5 (y + 1)}{2 {(y - 1)}^{2} (y + 1)} - \frac{3 (y - 1)}{2 (y + 1) {(y - 1)}^{1 + 1}}$

Step 4.6

Add $1$ and $1$ .

$\frac{5 (y + 1)}{2 {(y - 1)}^{2} (y + 1)} - \frac{3 (y - 1)}{2 (y + 1) {(y - 1)}^{2}}$

Step 4.7

Reorder the factors of $2 (y + 1) {(y - 1)}^{2}$ .

$\frac{5 (y + 1)}{2 {(y - 1)}^{2} (y + 1)} - \frac{3 (y - 1)}{2 {(y - 1)}^{2} (y + 1)}$

Step 5

Combine the numerators over the common denominator.

$\frac{5 (y + 1) - 3 (y - 1)}{2 {(y - 1)}^{2} (y + 1)}$

Step 6

Simplify the numerator.

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Step 6.1

Apply the distributive property.

$\frac{5 y + 5 \cdot 1 - 3 (y - 1)}{2 {(y - 1)}^{2} (y + 1)}$

Step 6.2

Multiply $5$ by $1$ .

$\frac{5 y + 5 - 3 (y - 1)}{2 {(y - 1)}^{2} (y + 1)}$

Step 6.3

Apply the distributive property.

$\frac{5 y + 5 - 3 y - 3 \cdot - 1}{2 {(y - 1)}^{2} (y + 1)}$

Step 6.4

Multiply $- 3$ by $- 1$ .

$\frac{5 y + 5 - 3 y + 3}{2 {(y - 1)}^{2} (y + 1)}$

Step 6.5

Subtract $3 y$ from $5 y$ .

$\frac{2 y + 5 + 3}{2 {(y - 1)}^{2} (y + 1)}$

Step 6.6

Add $5$ and $3$ .

$\frac{2 y + 8}{2 {(y - 1)}^{2} (y + 1)}$

Step 6.7

Factor $2$ out of $2 y + 8$ .

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Step 6.7.1

Factor $2$ out of $2 y$ .

$\frac{2 (y) + 8}{2 {(y - 1)}^{2} (y + 1)}$

Step 6.7.2

Factor $2$ out of $8$ .

$\frac{2 y + 2 \cdot 4}{2 {(y - 1)}^{2} (y + 1)}$

Step 6.7.3

Factor $2$ out of $2 y + 2 \cdot 4$ .

$\frac{2 (y + 4)}{2 {(y - 1)}^{2} (y + 1)}$

Step 7

Cancel the common factor of $2$ .

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Step 7.1

Cancel the common factor.

$\frac{2 (y + 4)}{2 {(y - 1)}^{2} (y + 1)}$

Step 7.2

Rewrite the expression.

$\frac{y + 4}{{(y - 1)}^{2} (y + 1)}$