Algebra Examples

$\frac{t^{2} + 3}{t^{4} - 16} + \frac{7}{16 - t^{4}}$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Simplify the denominator.

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Step 1.1.1.1

Rewrite $t^{4}$ as ${(t^{2})}^{2}$ .

$\frac{t^{2} + 3}{{(t^{2})}^{2} - 16} + \frac{7}{16 - t^{4}}$

Step 1.1.1.2

Rewrite $16$ as $4^{2}$ .

$\frac{t^{2} + 3}{{(t^{2})}^{2} - 4^{2}} + \frac{7}{16 - t^{4}}$

Step 1.1.1.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = t^{2}$ and $b = 4$ .

$\frac{t^{2} + 3}{(t^{2} + 4) (t^{2} - 4)} + \frac{7}{16 - t^{4}}$

Step 1.1.1.4

Simplify.

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Step 1.1.1.4.1

Rewrite $4$ as $2^{2}$ .

$\frac{t^{2} + 3}{(t^{2} + 4) (t^{2} - 2^{2})} + \frac{7}{16 - t^{4}}$

Step 1.1.1.4.2

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

$\frac{t^{2} + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{16 - t^{4}}$

Step 1.1.2

Simplify the denominator.

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Step 1.1.2.1

Rewrite $16$ as $4^{2}$ .

$\frac{t^{2} + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{4^{2} - t^{4}}$

Step 1.1.2.2

Rewrite $t^{4}$ as ${(t^{2})}^{2}$ .

$\frac{t^{2} + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{4^{2} - {(t^{2})}^{2}}$

Step 1.1.2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4$ and $b = t^{2}$ .

$\frac{t^{2} + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (4 - t^{2})}$

Step 1.1.2.4

Simplify.

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Step 1.1.2.4.1

Rewrite $4$ as $2^{2}$ .

$\frac{t^{2} + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2^{2} - t^{2})}$

Step 1.1.2.4.2

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

$\frac{t^{2} + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2 + t) (2 - t)}$

Step 1.2

Simplify terms.

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Step 1.2.1

Reorder terms.

$\frac{t^{2} + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(t^{2} + 4) (2 + t) (2 - t)}$

Step 1.2.2

Reorder terms.

$\frac{t^{2} + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(t^{2} + 4) (t + 2) (2 - t)}$

Step 1.2.3

Rewrite $2$ as $- 1 (- 2)$ .

$\frac{t^{2} + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(t^{2} + 4) (t + 2) (- 1 (- 2) - t)}$

Step 1.2.4

Factor $- 1$ out of $- t$ .

$\frac{t^{2} + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(t^{2} + 4) (t + 2) (- 1 (- 2) - (t))}$

Step 1.2.5

Factor $- 1$ out of $- 1 (- 2) - (t)$ .

$\frac{t^{2} + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(t^{2} + 4) (t + 2) (- 1 (- 2 + t))}$

Step 1.2.6

Simplify the expression.

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Step 1.2.6.1

Move a negative from the denominator of $\frac{7}{(t^{2} + 4) (t + 2) (- 1 (- 2 + t))}$ to the numerator.

$\frac{t^{2} + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{- 1 \cdot 7}{(t^{2} + 4) (t + 2) (- 2 + t)}$

Step 1.2.6.2

Reorder terms.

$\frac{t^{2} + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{- 1 \cdot 7}{(t^{2} + 4) (t + 2) (t - 2)}$

Step 1.2.7

Combine the numerators over the common denominator.

$\frac{t^{2} + 3 - 1 \cdot 7}{(t^{2} + 4) (t + 2) (t - 2)}$

Step 2

Simplify the numerator.

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Step 2.1

Multiply $- 1$ by $7$ .

$\frac{t^{2} + 3 - 7}{(t^{2} + 4) (t + 2) (t - 2)}$

Step 2.2

Subtract $7$ from $3$ .

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

Step 2.3

Rewrite $t^{2} - 4$ in a factored form.

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Step 2.3.1

Rewrite $4$ as $2^{2}$ .

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

Step 2.3.2

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

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

Step 3

Reduce the expression by cancelling the common factors.

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Step 3.1

Cancel the common factor of $t + 2$ .

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Step 3.1.1

Cancel the common factor.

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

Step 3.1.2

Rewrite the expression.

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

Step 3.2

Cancel the common factor of $t - 2$ .

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

$\frac{1}{t^{2} + 4}$