Trigonometry Examples

- 8 a^{4} \cdot (16 a^{3}) - 4 a^{2}

$- 8 a^{4} \cdot (16 a^{3}) - 4 a^{2}$

Step 1

Add parentheses.

- 8 (a^{4} \cdot (16 a^{3})) - 4 a^{2}

$- 8 (a^{4} \cdot (16 a^{3})) - 4 a^{2}$

Step 2

Let

u = a^{4} \cdot (16 a^{3})

$u = a^{4} \cdot (16 a^{3})$ . Substitute

u

$u$ for all occurrences of

a^{4} \cdot (16 a^{3})

$a^{4} \cdot (16 a^{3})$ .

- 8 u - 4 a^{2}

$- 8 u - 4 a^{2}$

Step 3

Factor

- 4

$- 4$ out of

- 8 u - 4 a^{2}

$- 8 u - 4 a^{2}$ .

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Step 3.1

Reorder

- 8 u

$- 8 u$ and

- 4 a^{2}

$- 4 a^{2}$ .

- 4 a^{2} - 8 u

$- 4 a^{2} - 8 u$

Step 3.2

Factor

- 4

$- 4$ out of

- 4 a^{2}

$- 4 a^{2}$ .

- 4 (a^{2}) - 8 u

$- 4 (a^{2}) - 8 u$

Step 3.3

Factor

- 4

$- 4$ out of

- 8 u

$- 8 u$ .

- 4 (a^{2}) - 4 (2 u)

$- 4 (a^{2}) - 4 (2 u)$

Step 3.4

Factor

- 4

$- 4$ out of

- 4 (a^{2}) - 4 (2 u)

$- 4 (a^{2}) - 4 (2 u)$ .

- 4 (a^{2} + 2 u)

$- 4 (a^{2} + 2 u)$

- 4 (a^{2} + 2 u)

$- 4 (a^{2} + 2 u)$

Step 4

Replace all occurrences of

u

$u$ with

a^{4} \cdot (16 a^{3})

$a^{4} \cdot (16 a^{3})$ .

- 4 (a^{2} + 2 (a^{4} \cdot (16 a^{3})))

$- 4 (a^{2} + 2 (a^{4} \cdot (16 a^{3})))$

Step 5

Simplify each term.

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Step 5.1

Multiply

a^{4}

$a^{4}$ by

a^{3}

$a^{3}$ by adding the exponents.

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Step 5.1.1

Move

a^{3}

$a^{3}$ .

- 4 (a^{2} + 2 (a^{3} a^{4} \cdot 16))

$- 4 (a^{2} + 2 (a^{3} a^{4} \cdot 16))$

Step 5.1.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

- 4 (a^{2} + 2 (a^{3 + 4} \cdot 16))

$- 4 (a^{2} + 2 (a^{3 + 4} \cdot 16))$

Step 5.1.3

Add

3

$3$ and

4

$4$ .

- 4 (a^{2} + 2 (a^{7} \cdot 16))

$- 4 (a^{2} + 2 (a^{7} \cdot 16))$

- 4 (a^{2} + 2 (a^{7} \cdot 16))

$- 4 (a^{2} + 2 (a^{7} \cdot 16))$

Step 5.2

Move

16

$16$ to the left of

a^{7}

$a^{7}$ .

- 4 (a^{2} + 2 (16 a^{7}))

$- 4 (a^{2} + 2 (16 a^{7}))$

Step 5.3

Multiply

16

$16$ by

2

$2$ .

- 4 (a^{2} + 32 a^{7})

$- 4 (a^{2} + 32 a^{7})$

- 4 (a^{2} + 32 a^{7})

$- 4 (a^{2} + 32 a^{7})$

Step 6

Factor.

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Step 6.1

Factor

a^{2}

$a^{2}$ out of

a^{2} + 32 a^{7}

$a^{2} + 32 a^{7}$ .

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Step 6.1.1

Multiply by

1

$1$ .

- 4 (a^{2} \cdot 1 + 32 a^{7})

$- 4 (a^{2} \cdot 1 + 32 a^{7})$

Step 6.1.2

Factor

a^{2}

$a^{2}$ out of

32 a^{7}

$32 a^{7}$ .

- 4 (a^{2} \cdot 1 + a^{2} (32 a^{5}))

$- 4 (a^{2} \cdot 1 + a^{2} (32 a^{5}))$

Step 6.1.3

Factor

a^{2}

$a^{2}$ out of

a^{2} \cdot 1 + a^{2} (32 a^{5})

$a^{2} \cdot 1 + a^{2} (32 a^{5})$ .

- 4 (a^{2} (1 + 32 a^{5}))

$- 4 (a^{2} (1 + 32 a^{5}))$

- 4 (a^{2} (1 + 32 a^{5}))

$- 4 (a^{2} (1 + 32 a^{5}))$

Step 6.2

Remove unnecessary parentheses.

- 4 a^{2} (1 + 32 a^{5})

$- 4 a^{2} (1 + 32 a^{5})$

$- 4 a^{2} (1 + 32 a^{5})$

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$