Pre-Algebra Examples

$y^{2} - 49 \div \frac{y^{2} + 5 y - 14}{y^{2} + 7}$

Step 1

Simplify each term.

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Step 1.1

To divide by a fraction, multiply by its reciprocal.

$y^{2} - (49 \frac{y^{2} + 7}{y^{2} + 5 y - 14})$

Step 1.2

Factor $y^{2} + 5 y - 14$ using the AC method.

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Step 1.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 14$ and whose sum is $5$ .

$- 2, 7$

Step 1.2.2

Write the factored form using these integers.

$y^{2} - (49 \frac{y^{2} + 7}{(y - 2) (y + 7)})$

Step 1.3

Combine $49$ and $\frac{y^{2} + 7}{(y - 2) (y + 7)}$ .

$y^{2} - \frac{49 (y^{2} + 7)}{(y - 2) (y + 7)}$

Step 2

To write $y^{2}$ as a fraction with a common denominator, multiply by $\frac{(y - 2) (y + 7)}{(y - 2) (y + 7)}$ .

$y^{2} \cdot \frac{(y - 2) (y + 7)}{(y - 2) (y + 7)} - \frac{49 (y^{2} + 7)}{(y - 2) (y + 7)}$

Step 3

Simplify terms.

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Step 3.1

Combine $y^{2}$ and $\frac{(y - 2) (y + 7)}{(y - 2) (y + 7)}$ .

$\frac{y^{2} ((y - 2) (y + 7))}{(y - 2) (y + 7)} - \frac{49 (y^{2} + 7)}{(y - 2) (y + 7)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{y^{2} ((y - 2) (y + 7)) - 49 (y^{2} + 7)}{(y - 2) (y + 7)}$

Step 4

Simplify the numerator.

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Step 4.1

Expand $(y - 2) (y + 7)$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

$\frac{y^{2} (y (y + 7) - 2 (y + 7)) - 49 (y^{2} + 7)}{(y - 2) (y + 7)}$

Step 4.1.2

Apply the distributive property.

$\frac{y^{2} (y \cdot y + y \cdot 7 - 2 (y + 7)) - 49 (y^{2} + 7)}{(y - 2) (y + 7)}$

Step 4.1.3

Apply the distributive property.

$\frac{y^{2} (y \cdot y + y \cdot 7 - 2 y - 2 \cdot 7) - 49 (y^{2} + 7)}{(y - 2) (y + 7)}$

Step 4.2

Simplify and combine like terms.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Multiply $y$ by $y$ .

$\frac{y^{2} (y^{2} + y \cdot 7 - 2 y - 2 \cdot 7) - 49 (y^{2} + 7)}{(y - 2) (y + 7)}$

Step 4.2.1.2

Move $7$ to the left of $y$ .

$\frac{y^{2} (y^{2} + 7 \cdot y - 2 y - 2 \cdot 7) - 49 (y^{2} + 7)}{(y - 2) (y + 7)}$

Step 4.2.1.3

Multiply $- 2$ by $7$ .

$\frac{y^{2} (y^{2} + 7 y - 2 y - 14) - 49 (y^{2} + 7)}{(y - 2) (y + 7)}$

Step 4.2.2

Subtract $2 y$ from $7 y$ .

$\frac{y^{2} (y^{2} + 5 y - 14) - 49 (y^{2} + 7)}{(y - 2) (y + 7)}$

Step 4.3

Apply the distributive property.

$\frac{y^{2} y^{2} + y^{2} (5 y) + y^{2} \cdot - 14 - 49 (y^{2} + 7)}{(y - 2) (y + 7)}$

Step 4.4

Simplify.

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Step 4.4.1

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

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Step 4.4.1.1

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

$\frac{y^{2 + 2} + y^{2} (5 y) + y^{2} \cdot - 14 - 49 (y^{2} + 7)}{(y - 2) (y + 7)}$

Step 4.4.1.2

Add $2$ and $2$ .

$\frac{y^{4} + y^{2} (5 y) + y^{2} \cdot - 14 - 49 (y^{2} + 7)}{(y - 2) (y + 7)}$

Step 4.4.2

Rewrite using the commutative property of multiplication.

$\frac{y^{4} + 5 y^{2} y + y^{2} \cdot - 14 - 49 (y^{2} + 7)}{(y - 2) (y + 7)}$

Step 4.4.3

Move $- 14$ to the left of $y^{2}$ .

$\frac{y^{4} + 5 y^{2} y - 14 \cdot y^{2} - 49 (y^{2} + 7)}{(y - 2) (y + 7)}$

Step 4.5

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

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Step 4.5.1

Move $y$ .

$\frac{y^{4} + 5 (y \cdot y^{2}) - 14 \cdot y^{2} - 49 (y^{2} + 7)}{(y - 2) (y + 7)}$

Step 4.5.2

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

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Step 4.5.2.1

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

$\frac{y^{4} + 5 (y^{1} y^{2}) - 14 \cdot y^{2} - 49 (y^{2} + 7)}{(y - 2) (y + 7)}$

Step 4.5.2.2

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

$\frac{y^{4} + 5 y^{1 + 2} - 14 \cdot y^{2} - 49 (y^{2} + 7)}{(y - 2) (y + 7)}$

Step 4.5.3

Add $1$ and $2$ .

$\frac{y^{4} + 5 y^{3} - 14 \cdot y^{2} - 49 (y^{2} + 7)}{(y - 2) (y + 7)}$

$\frac{y^{4} + 5 y^{3} - 14 y^{2} - 49 (y^{2} + 7)}{(y - 2) (y + 7)}$

Step 4.6

Apply the distributive property.

$\frac{y^{4} + 5 y^{3} - 14 y^{2} - 49 y^{2} - 49 \cdot 7}{(y - 2) (y + 7)}$

Step 4.7

Multiply $- 49$ by $7$ .

$\frac{y^{4} + 5 y^{3} - 14 y^{2} - 49 y^{2} - 343}{(y - 2) (y + 7)}$

Step 4.8

Subtract $49 y^{2}$ from $- 14 y^{2}$ .

$\frac{y^{4} + 5 y^{3} - 63 y^{2} - 343}{(y - 2) (y + 7)}$