Basic Math Examples

$\frac{9 y}{y^{2} - 12 + 36} - \frac{54}{y^{2} - 12 y + 36}$

Step 1

Simplify each term.

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Step 1.1

Add $- 12$ and $36$ .

$\frac{9 y}{y^{2} + 24} - \frac{54}{y^{2} - 12 y + 36}$

Step 1.2

Factor using the perfect square rule.

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Step 1.2.1

Rewrite $36$ as $6^{2}$ .

$\frac{9 y}{y^{2} + 24} - \frac{54}{y^{2} - 12 y + 6^{2}}$

Step 1.2.2

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

$12 y = 2 \cdot y \cdot 6$

Step 1.2.3

Rewrite the polynomial.

$\frac{9 y}{y^{2} + 24} - \frac{54}{y^{2} - 2 \cdot y \cdot 6 + 6^{2}}$

Step 1.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = y$ and $b = 6$ .

$\frac{9 y}{y^{2} + 24} - \frac{54}{{(y - 6)}^{2}}$

Step 2

To write $\frac{9 y}{y^{2} + 24}$ as a fraction with a common denominator, multiply by $\frac{{(y - 6)}^{2}}{{(y - 6)}^{2}}$ .

$\frac{9 y}{y^{2} + 24} \cdot \frac{{(y - 6)}^{2}}{{(y - 6)}^{2}} - \frac{54}{{(y - 6)}^{2}}$

Step 3

To write $- \frac{54}{{(y - 6)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{y^{2} + 24}{y^{2} + 24}$ .

$\frac{9 y}{y^{2} + 24} \cdot \frac{{(y - 6)}^{2}}{{(y - 6)}^{2}} - \frac{54}{{(y - 6)}^{2}} \cdot \frac{y^{2} + 24}{y^{2} + 24}$

Step 4

Write each expression with a common denominator of $(y^{2} + 24) {(y - 6)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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Step 4.1

Multiply $\frac{9 y}{y^{2} + 24}$ by $\frac{{(y - 6)}^{2}}{{(y - 6)}^{2}}$ .

$\frac{9 y {(y - 6)}^{2}}{(y^{2} + 24) {(y - 6)}^{2}} - \frac{54}{{(y - 6)}^{2}} \cdot \frac{y^{2} + 24}{y^{2} + 24}$

Step 4.2

Multiply $\frac{54}{{(y - 6)}^{2}}$ by $\frac{y^{2} + 24}{y^{2} + 24}$ .

$\frac{9 y {(y - 6)}^{2}}{(y^{2} + 24) {(y - 6)}^{2}} - \frac{54 (y^{2} + 24)}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 4.3

Reorder the factors of $(y^{2} + 24) {(y - 6)}^{2}$ .

$\frac{9 y {(y - 6)}^{2}}{{(y - 6)}^{2} (y^{2} + 24)} - \frac{54 (y^{2} + 24)}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 5

Combine the numerators over the common denominator.

$\frac{9 y {(y - 6)}^{2} - 54 (y^{2} + 24)}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 6

Simplify the numerator.

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Step 6.1

Factor $9$ out of $9 y {(y - 6)}^{2} - 54 (y^{2} + 24)$ .

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Step 6.1.1

Factor $9$ out of $9 y {(y - 6)}^{2}$ .

$\frac{9 (y {(y - 6)}^{2}) - 54 (y^{2} + 24)}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 6.1.2

Factor $9$ out of $- 54 (y^{2} + 24)$ .

$\frac{9 (y {(y - 6)}^{2}) + 9 (- 6 (y^{2} + 24))}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 6.1.3

Factor $9$ out of $9 (y {(y - 6)}^{2}) + 9 (- 6 (y^{2} + 24))$ .

$\frac{9 (y {(y - 6)}^{2} - 6 (y^{2} + 24))}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 6.2

Rewrite ${(y - 6)}^{2}$ as $(y - 6) (y - 6)$ .

$\frac{9 (y ((y - 6) (y - 6)) - 6 (y^{2} + 24))}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 6.3

Expand $(y - 6) (y - 6)$ using the FOIL Method.

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Step 6.3.1

Apply the distributive property.

$\frac{9 (y (y (y - 6) - 6 (y - 6)) - 6 (y^{2} + 24))}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 6.3.2

Apply the distributive property.

$\frac{9 (y (y \cdot y + y \cdot - 6 - 6 (y - 6)) - 6 (y^{2} + 24))}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 6.3.3

Apply the distributive property.

$\frac{9 (y (y \cdot y + y \cdot - 6 - 6 y - 6 \cdot - 6) - 6 (y^{2} + 24))}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 6.4

Simplify and combine like terms.

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Step 6.4.1

Simplify each term.

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Step 6.4.1.1

Multiply $y$ by $y$ .

$\frac{9 (y (y^{2} + y \cdot - 6 - 6 y - 6 \cdot - 6) - 6 (y^{2} + 24))}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 6.4.1.2

Move $- 6$ to the left of $y$ .

$\frac{9 (y (y^{2} - 6 \cdot y - 6 y - 6 \cdot - 6) - 6 (y^{2} + 24))}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 6.4.1.3

Multiply $- 6$ by $- 6$ .

$\frac{9 (y (y^{2} - 6 y - 6 y + 36) - 6 (y^{2} + 24))}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 6.4.2

Subtract $6 y$ from $- 6 y$ .

$\frac{9 (y (y^{2} - 12 y + 36) - 6 (y^{2} + 24))}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 6.5

Apply the distributive property.

$\frac{9 (y \cdot y^{2} + y (- 12 y) + y \cdot 36 - 6 (y^{2} + 24))}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 6.6

Simplify.

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Step 6.6.1

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

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Step 6.6.1.1

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

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Step 6.6.1.1.1

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

$\frac{9 (y^{1} y^{2} + y (- 12 y) + y \cdot 36 - 6 (y^{2} + 24))}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 6.6.1.1.2

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

$\frac{9 (y^{1 + 2} + y (- 12 y) + y \cdot 36 - 6 (y^{2} + 24))}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 6.6.1.2

Add $1$ and $2$ .

$\frac{9 (y^{3} + y (- 12 y) + y \cdot 36 - 6 (y^{2} + 24))}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 6.6.2

Rewrite using the commutative property of multiplication.

$\frac{9 (y^{3} - 12 y \cdot y + y \cdot 36 - 6 (y^{2} + 24))}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 6.6.3

Move $36$ to the left of $y$ .

$\frac{9 (y^{3} - 12 y \cdot y + 36 \cdot y - 6 (y^{2} + 24))}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 6.7

Multiply $y$ by $y$ by adding the exponents.

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Step 6.7.1

Move $y$ .

$\frac{9 (y^{3} - 12 (y \cdot y) + 36 \cdot y - 6 (y^{2} + 24))}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 6.7.2

Multiply $y$ by $y$ .

$\frac{9 (y^{3} - 12 y^{2} + 36 \cdot y - 6 (y^{2} + 24))}{{(y - 6)}^{2} (y^{2} + 24)}$

$\frac{9 (y^{3} - 12 y^{2} + 36 y - 6 (y^{2} + 24))}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 6.8

Apply the distributive property.

$\frac{9 (y^{3} - 12 y^{2} + 36 y - 6 y^{2} - 6 \cdot 24)}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 6.9

Multiply $- 6$ by $24$ .

$\frac{9 (y^{3} - 12 y^{2} + 36 y - 6 y^{2} - 144)}{{(y - 6)}^{2} (y^{2} + 24)}$

Step 6.10

Subtract $6 y^{2}$ from $- 12 y^{2}$ .

$\frac{9 (y^{3} - 18 y^{2} + 36 y - 144)}{{(y - 6)}^{2} (y^{2} + 24)}$