Basic Math Examples

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

Step 1

Simplify each term.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 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 $b = 4$ .

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

Step 1.1.4

Simplify.

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Step 1.1.4.1

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

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

Step 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 $b = 2$ .

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

Step 1.2

Simplify the denominator.

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 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 $b = t^{2}$ .

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

Step 1.2.4

Simplify.

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Step 1.2.4.1

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

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

Step 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 $b = t$ .

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

Step 2

To write $t^{2}$ as a fraction with a common denominator, multiply by $\frac{(t^{2} + 4) (t + 2) (t - 2)}{(t^{2} + 4) (t + 2) (t - 2)}$ .

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

Step 3

Simplify terms.

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Step 3.1

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 4

Simplify the numerator.

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Step 4.1

Expand $(t^{2} + 4) (t + 2)$ using the FOIL Method.

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Step 4.1.1

Apply the distributive property.

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

Step 4.1.2

Apply the distributive property.

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

Step 4.1.3

Apply the distributive property.

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

Step 4.2

Simplify each term.

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Step 4.2.1

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

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Step 4.2.1.1

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

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Step 4.2.1.1.1

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

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

Step 4.2.1.1.2

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

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

Step 4.2.1.2

Add $2$ and $1$ .

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

Step 4.2.2

Move $2$ to the left of $t^{2}$ .

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

Step 4.2.3

Multiply $4$ by $2$ .

$\frac{t^{2} ((t^{3} + 2 t^{2} + 4 t + 8) (t - 2)) + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2 + t) (2 - t)}$

Step 4.3

Expand $(t^{3} + 2 t^{2} + 4 t + 8) (t - 2)$ by multiplying each term in the first expression by each term in the second expression.

$\frac{t^{2} (t^{3} t + t^{3} \cdot - 2 + 2 t^{2} t + 2 t^{2} \cdot - 2 + 4 t \cdot t + 4 t \cdot - 2 + 8 t + 8 \cdot - 2) + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2 + t) (2 - t)}$

Step 4.4

Simplify each term.

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Step 4.4.1

Multiply $t^{3}$ by $t$ by adding the exponents.

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Step 4.4.1.1

Multiply $t^{3}$ by $t$ .

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Step 4.4.1.1.1

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

$\frac{t^{2} (t^{3} t^{1} + t^{3} \cdot - 2 + 2 t^{2} t + 2 t^{2} \cdot - 2 + 4 t \cdot t + 4 t \cdot - 2 + 8 t + 8 \cdot - 2) + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2 + t) (2 - t)}$

Step 4.4.1.1.2

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

$\frac{t^{2} (t^{3 + 1} + t^{3} \cdot - 2 + 2 t^{2} t + 2 t^{2} \cdot - 2 + 4 t \cdot t + 4 t \cdot - 2 + 8 t + 8 \cdot - 2) + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2 + t) (2 - t)}$

Step 4.4.1.2

Add $3$ and $1$ .

$\frac{t^{2} (t^{4} + t^{3} \cdot - 2 + 2 t^{2} t + 2 t^{2} \cdot - 2 + 4 t \cdot t + 4 t \cdot - 2 + 8 t + 8 \cdot - 2) + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2 + t) (2 - t)}$

Step 4.4.2

Move $- 2$ to the left of $t^{3}$ .

$\frac{t^{2} (t^{4} - 2 \cdot t^{3} + 2 t^{2} t + 2 t^{2} \cdot - 2 + 4 t \cdot t + 4 t \cdot - 2 + 8 t + 8 \cdot - 2) + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2 + t) (2 - t)}$

Step 4.4.3

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

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Step 4.4.3.1

Move $t$ .

$\frac{t^{2} (t^{4} - 2 t^{3} + 2 (t \cdot t^{2}) + 2 t^{2} \cdot - 2 + 4 t \cdot t + 4 t \cdot - 2 + 8 t + 8 \cdot - 2) + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2 + t) (2 - t)}$

Step 4.4.3.2

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

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Step 4.4.3.2.1

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

$\frac{t^{2} (t^{4} - 2 t^{3} + 2 (t^{1} t^{2}) + 2 t^{2} \cdot - 2 + 4 t \cdot t + 4 t \cdot - 2 + 8 t + 8 \cdot - 2) + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2 + t) (2 - t)}$

Step 4.4.3.2.2

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

$\frac{t^{2} (t^{4} - 2 t^{3} + 2 t^{1 + 2} + 2 t^{2} \cdot - 2 + 4 t \cdot t + 4 t \cdot - 2 + 8 t + 8 \cdot - 2) + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2 + t) (2 - t)}$

Step 4.4.3.3

Add $1$ and $2$ .

$\frac{t^{2} (t^{4} - 2 t^{3} + 2 t^{3} + 2 t^{2} \cdot - 2 + 4 t \cdot t + 4 t \cdot - 2 + 8 t + 8 \cdot - 2) + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2 + t) (2 - t)}$

Step 4.4.4

Multiply $- 2$ by $2$ .

$\frac{t^{2} (t^{4} - 2 t^{3} + 2 t^{3} - 4 t^{2} + 4 t \cdot t + 4 t \cdot - 2 + 8 t + 8 \cdot - 2) + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2 + t) (2 - t)}$

Step 4.4.5

Multiply $t$ by $t$ by adding the exponents.

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Step 4.4.5.1

Move $t$ .

$\frac{t^{2} (t^{4} - 2 t^{3} + 2 t^{3} - 4 t^{2} + 4 (t \cdot t) + 4 t \cdot - 2 + 8 t + 8 \cdot - 2) + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2 + t) (2 - t)}$

Step 4.4.5.2

Multiply $t$ by $t$ .

$\frac{t^{2} (t^{4} - 2 t^{3} + 2 t^{3} - 4 t^{2} + 4 t^{2} + 4 t \cdot - 2 + 8 t + 8 \cdot - 2) + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2 + t) (2 - t)}$

Step 4.4.6

Multiply $- 2$ by $4$ .

$\frac{t^{2} (t^{4} - 2 t^{3} + 2 t^{3} - 4 t^{2} + 4 t^{2} - 8 t + 8 t + 8 \cdot - 2) + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2 + t) (2 - t)}$

Step 4.4.7

Multiply $8$ by $- 2$ .

$\frac{t^{2} (t^{4} - 2 t^{3} + 2 t^{3} - 4 t^{2} + 4 t^{2} - 8 t + 8 t - 16) + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2 + t) (2 - t)}$

Step 4.5

Combine the opposite terms in $t^{4} - 2 t^{3} + 2 t^{3} - 4 t^{2} + 4 t^{2} - 8 t + 8 t - 16$ .

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Step 4.5.1

Add $- 2 t^{3}$ and $2 t^{3}$ .

$\frac{t^{2} (t^{4} + 0 - 4 t^{2} + 4 t^{2} - 8 t + 8 t - 16) + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2 + t) (2 - t)}$

Step 4.5.2

Add $t^{4}$ and $0$ .

$\frac{t^{2} (t^{4} - 4 t^{2} + 4 t^{2} - 8 t + 8 t - 16) + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2 + t) (2 - t)}$

Step 4.5.3

Add $- 4 t^{2}$ and $4 t^{2}$ .

$\frac{t^{2} (t^{4} + 0 - 8 t + 8 t - 16) + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2 + t) (2 - t)}$

Step 4.5.4

Add $t^{4}$ and $0$ .

$\frac{t^{2} (t^{4} - 8 t + 8 t - 16) + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2 + t) (2 - t)}$

Step 4.5.5

Add $- 8 t$ and $8 t$ .

$\frac{t^{2} (t^{4} + 0 - 16) + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2 + t) (2 - t)}$

Step 4.5.6

Add $t^{4}$ and $0$ .

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

Step 4.6

Apply the distributive property.

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

Step 4.7

Multiply $t^{2}$ by $t^{4}$ by adding the exponents.

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Step 4.7.1

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

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

Step 4.7.2

Add $2$ and $4$ .

$\frac{t^{6} + t^{2} \cdot - 16 + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2 + t) (2 - t)}$

Step 4.8

Move $- 16$ to the left of $t^{2}$ .

$\frac{t^{6} - 16 t^{2} + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(4 + t^{2}) (2 + t) (2 - t)}$

Step 5

Simplify terms.

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Step 5.1

Reorder terms.

$\frac{t^{6} - 16 t^{2} + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(t^{2} + 4) (2 + t) (2 - t)}$

Step 5.2

Reorder terms.

$\frac{t^{6} - 16 t^{2} + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(t^{2} + 4) (t + 2) (2 - t)}$

Step 5.3

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

$\frac{t^{6} - 16 t^{2} + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(t^{2} + 4) (t + 2) (- 1 (- 2) - t)}$

Step 5.4

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

$\frac{t^{6} - 16 t^{2} + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(t^{2} + 4) (t + 2) (- 1 (- 2) - (t))}$

Step 5.5

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

$\frac{t^{6} - 16 t^{2} + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{7}{(t^{2} + 4) (t + 2) (- 1 (- 2 + t))}$

Step 5.6

Simplify the expression.

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Step 5.6.1

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

$\frac{t^{6} - 16 t^{2} + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{- 1 \cdot 7}{(t^{2} + 4) (t + 2) (- 2 + t)}$

Step 5.6.2

Reorder terms.

$\frac{t^{6} - 16 t^{2} + 3}{(t^{2} + 4) (t + 2) (t - 2)} + \frac{- 1 \cdot 7}{(t^{2} + 4) (t + 2) (t - 2)}$

Step 5.7

Combine the numerators over the common denominator.

$\frac{t^{6} - 16 t^{2} + 3 - 1 \cdot 7}{(t^{2} + 4) (t + 2) (t - 2)}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply $- 1$ by $7$ .

$\frac{t^{6} - 16 t^{2} + 3 - 7}{(t^{2} + 4) (t + 2) (t - 2)}$

Step 6.2

Subtract $7$ from $3$ .

$\frac{t^{6} - 16 t^{2} - 4}{(t^{2} + 4) (t + 2) (t - 2)}$