Pre-Algebra Examples

\frac{t^{3} - 4 t}{t - t^{4}} \cdot \frac{t^{4} - t}{4 t - t^{3}}

$\frac{t^{3} - 4 t}{t - t^{4}} \cdot \frac{t^{4} - t}{4 t - t^{3}}$

Step 1

Simplify the numerator.

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Step 1.1

Factor

t

$t$ out of

t^{3} - 4 t

$t^{3} - 4 t$ .

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Step 1.1.1

Factor

t

$t$ out of

t^{3}

$t^{3}$ .

\frac{t \cdot t^{2} - 4 t}{t - t^{4}} \cdot \frac{t^{4} - t}{4 t - t^{3}}

$\frac{t \cdot t^{2} - 4 t}{t - t^{4}} \cdot \frac{t^{4} - t}{4 t - t^{3}}$

Step 1.1.2

Factor

t

$t$ out of

- 4 t

$- 4 t$ .

\frac{t \cdot t^{2} + t \cdot - 4}{t - t^{4}} \cdot \frac{t^{4} - t}{4 t - t^{3}}

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

Step 1.1.3

Factor

t

$t$ out of

t \cdot t^{2} + t \cdot - 4

$t \cdot t^{2} + t \cdot - 4$ .

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

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

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

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

Step 1.2

Rewrite

4

$4$ as

2^{2}

$2^{2}$ .

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

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

Step 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

$a = t$ and

b = 2

$b = 2$ .

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

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

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

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

Step 2

Simplify the denominator.

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Step 2.1

Factor

t

$t$ out of

t - t^{4}

$t - t^{4}$ .

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Step 2.1.1

Raise

t

$t$ to the power of

1

$1$ .

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

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

Step 2.1.2

Factor

t

$t$ out of

t^{1}

$t^{1}$ .

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

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

Step 2.1.3

Factor

t

$t$ out of

- t^{4}

$- t^{4}$ .

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

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

Step 2.1.4

Factor

t

$t$ out of

t \cdot 1 + t (- t^{3})

$t \cdot 1 + t (- t^{3})$ .

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

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

Step 2.2

Rewrite

$1$ as

$1^{3}$ .

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

Step 2.3

Since both terms are perfect cubes, factor using the difference of cubes formula,

$a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where

$a = 1$ and

$b = t$ .

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

Step 2.4

Simplify.

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Step 2.4.1

One to any power is one.

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

Step 2.4.2

Multiply

$t$ by

$1$ .

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

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

Step 3

Simplify the numerator.

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Step 3.1

Factor

$t$ out of

$t^{4} - t$ .

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Step 3.1.1

Factor

$t$ out of

$t^{4}$ .

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

Step 3.1.2

Factor

$t$ out of

$- t$ .

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

Step 3.1.3

Factor

$t$ out of

$t \cdot t^{3} + t \cdot - 1$ .

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

Step 3.2

Rewrite

$1$ as

$1^{3}$ .

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

Step 3.3

Since both terms are perfect cubes, factor using the difference of cubes formula,

$a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where

$a = t$ and

$b = 1$ .

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

Step 3.4

Simplify.

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Step 3.4.1

Multiply

$t$ by

$1$ .

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

Step 3.4.2

One to any power is one.

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

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

Step 4

Simplify the denominator.

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Step 4.1

Factor

$t$ out of

$4 t - t^{3}$ .

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Step 4.1.1

Factor

$t$ out of

$4 t$ .

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

Step 4.1.2

Factor

$t$ out of

$- t^{3}$ .

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

Step 4.1.3

Factor

$t$ out of

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

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

Step 4.2

Rewrite

$4$ as

$2^{2}$ .

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

Step 4.3

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

$b = t$ .

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

Step 5

Combine.

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

Step 6

Multiply

$t$ by

$t$ by adding the exponents.

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Step 6.1

Move

$t$ .

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

Step 6.2

Multiply

$t$ by

$t$ .

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

Step 7

Multiply

$t$ by

$t$ by adding the exponents.

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Step 7.1

Move

$t$ .

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

Step 7.2

Multiply

$t$ by

$t$ .

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

Step 8

Cancel the common factor of

$t^{2}$ .

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Step 8.1

Cancel the common factor.

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

Step 8.2

Rewrite the expression.

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

Step 9

Cancel the common factor of

$t + 2$ and

$2 + t$ .

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Step 9.1

Reorder terms.

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

Step 9.2

Cancel the common factor.

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

Step 9.3

Rewrite the expression.

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

Step 10

Cancel the common factor of

$t - 2$ and

$2 - t$ .

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Step 10.1

Factor

$- 1$ out of

$t$ .

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

Step 10.2

Rewrite

$- 2$ as

$- 1 (2)$ .

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

Step 10.3

Factor

$- 1$ out of

$- 1 (- t) - 1 (2)$ .

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

Step 10.4

Reorder terms.

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

Step 10.5

Cancel the common factor.

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

Step 10.6

Rewrite the expression.

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

Step 11

Cancel the common factor of

$t - 1$ and

$1 - t$ .

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Step 11.1

Factor

$- 1$ out of

$t$ .

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

Step 11.2

Rewrite

$- 1$ as

$- 1 (1)$ .

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

Step 11.3

Factor

$- 1$ out of

$- 1 (- t) - 1 (1)$ .

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

Step 11.4

Reorder terms.

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

Step 11.5

Cancel the common factor.

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

Step 11.6

Rewrite the expression.

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

Step 12

Cancel the common factor of

$t^{2} + t + 1$ and

$1 + t + t^{2}$ .

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Step 12.1

Reorder terms.

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

Step 12.2

Cancel the common factor.

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

Step 12.3

Divide

$- 1 \cdot (- 1)$ by

$1$ .

$- 1 \cdot (- 1)$

Step 13

Simplify the expression.

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Step 13.1

Rewrite

$- 1 \cdot (- 1)$ as

$- (- 1)$ .

$- (- 1)$

Step 13.2

Multiply

$- 1$ by

$- 1$ .

$1$