Algebra Examples

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

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

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}

$t^{4}$ as

{(t^{2})}^{2}

${(t^{2})}^{2}$ .

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

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

Step 1.1.1.2

Rewrite

16

$16$ as

4^{2}

$4^{2}$ .

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

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

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)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = t^{2}

$a = t^{2}$ and

b = 4

$b = 4$ .

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

$4$ as

2^{2}

$2^{2}$ .

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

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

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)

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = t

$a = t$ and

b = 2

$b = 2$ .

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

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

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

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

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

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

Step 1.1.2

Simplify the denominator.

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Step 1.1.2.1

Rewrite

16

$16$ as

4^{2}

$4^{2}$ .

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

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

Step 1.1.2.2

Rewrite

t^{4}

$t^{4}$ as

{(t^{2})}^{2}

${(t^{2})}^{2}$ .

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

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = 4

$a = 4$ and

b = t^{2}

$b = t^{2}$ .

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

$4$ as

2^{2}

$2^{2}$ .

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

$a^{2} - b^{2} = (a + b) (a - b)$ where

a = 2

$a = 2$ and

b = t

$b = t$ .

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

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

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

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

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

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

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

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

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)}

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

Step 1.2.3

Rewrite

2

$2$ as

- 1 (- 2)

$- 1 (- 2)$ .

\frac{t^{2} + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(t^{2} + 4) (t + 2) (- 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 1.2.4

Factor

- 1

$- 1$ out of

- t

$- t$ .

\frac{t^{2} + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(t^{2} + 4) (t + 2) (- 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 1.2.5

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.

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