Basic Math Examples

- 2 {(- 2 a^{9} b^{4})}^{3} {(3 a^{9} b)}^{2}

$- 2 {(- 2 a^{9} b^{4})}^{3} {(3 a^{9} b)}^{2}$

Step 1

Use the power rule

{(a b)}^{n} = a^{n} b^{n}

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 1.1

Apply the product rule to

- 2 a^{9} b^{4}

$- 2 a^{9} b^{4}$ .

- 2 ({(- 2 a^{9})}^{3} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}

$- 2 ({(- 2 a^{9})}^{3} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}$

Step 1.2

Apply the product rule to

- 2 a^{9}

$- 2 a^{9}$ .

- 2 ({(- 2)}^{3} {(a^{9})}^{3} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}

$- 2 ({(- 2)}^{3} {(a^{9})}^{3} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}$

- 2 ({(- 2)}^{3} {(a^{9})}^{3} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}

$- 2 ({(- 2)}^{3} {(a^{9})}^{3} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}$

Step 2

Multiply

- 2

$- 2$ by

{(- 2)}^{3}

${(- 2)}^{3}$ by adding the exponents.

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Step 2.1

Move

{(- 2)}^{3}

${(- 2)}^{3}$ .

{(- 2)}^{3} \cdot - 2 ({(a^{9})}^{3} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}

${(- 2)}^{3} \cdot - 2 ({(a^{9})}^{3} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}$

Step 2.2

Multiply

{(- 2)}^{3}

${(- 2)}^{3}$ by

- 2

$- 2$ .

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Step 2.2.1

Raise

- 2

$- 2$ to the power of

1

$1$ .

{(- 2)}^{3} \cdot {(- 2)}^{1} ({(a^{9})}^{3} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}

${(- 2)}^{3} \cdot {(- 2)}^{1} ({(a^{9})}^{3} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}$

Step 2.2.2

Use the power rule

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

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

{(- 2)}^{3 + 1} ({(a^{9})}^{3} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}

${(- 2)}^{3 + 1} ({(a^{9})}^{3} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}$

{(- 2)}^{3 + 1} ({(a^{9})}^{3} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}

${(- 2)}^{3 + 1} ({(a^{9})}^{3} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}$

Step 2.3

Add

3

$3$ and

1

$1$ .

{(- 2)}^{4} ({(a^{9})}^{3} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}

${(- 2)}^{4} ({(a^{9})}^{3} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}$

{(- 2)}^{4} ({(a^{9})}^{3} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}

${(- 2)}^{4} ({(a^{9})}^{3} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}$

Step 3

Raise

- 2

$- 2$ to the power of

4

$4$ .

16 ({(a^{9})}^{3} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}

$16 ({(a^{9})}^{3} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}$

Step 4

Multiply the exponents in

{(a^{9})}^{3}

${(a^{9})}^{3}$ .

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Step 4.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

16 (a^{9 \cdot 3} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}

$16 (a^{9 \cdot 3} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}$

Step 4.2

Multiply

9

$9$ by

3

$3$ .

16 (a^{27} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}

$16 (a^{27} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}$

16 (a^{27} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}

$16 (a^{27} {(b^{4})}^{3}) {(3 a^{9} b)}^{2}$

Step 5

Multiply the exponents in

{(b^{4})}^{3}

${(b^{4})}^{3}$ .

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Step 5.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

16 (a^{27} b^{4 \cdot 3}) {(3 a^{9} b)}^{2}

$16 (a^{27} b^{4 \cdot 3}) {(3 a^{9} b)}^{2}$

Step 5.2

Multiply

4

$4$ by

3

$3$ .

16 (a^{27} b^{12}) {(3 a^{9} b)}^{2}

$16 (a^{27} b^{12}) {(3 a^{9} b)}^{2}$

16 (a^{27} b^{12}) {(3 a^{9} b)}^{2}

$16 (a^{27} b^{12}) {(3 a^{9} b)}^{2}$

Step 6

Use the power rule

{(a b)}^{n} = a^{n} b^{n}

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 6.1

Apply the product rule to

3 a^{9} b

$3 a^{9} b$ .

16 a^{27} b^{12} ({(3 a^{9})}^{2} b^{2})

$16 a^{27} b^{12} ({(3 a^{9})}^{2} b^{2})$

Step 6.2

Apply the product rule to

3 a^{9}

$3 a^{9}$ .

16 a^{27} b^{12} (3^{2} {(a^{9})}^{2} b^{2})

$16 a^{27} b^{12} (3^{2} {(a^{9})}^{2} b^{2})$

16 a^{27} b^{12} (3^{2} {(a^{9})}^{2} b^{2})

$16 a^{27} b^{12} (3^{2} {(a^{9})}^{2} b^{2})$

Step 7

Multiply

b^{12}

$b^{12}$ by

b^{2}

$b^{2}$ by adding the exponents.

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Step 7.1

Move

b^{2}

$b^{2}$ .

16 a^{27} (b^{2} b^{12}) (3^{2} {(a^{9})}^{2})

$16 a^{27} (b^{2} b^{12}) (3^{2} {(a^{9})}^{2})$

Step 7.2

Use the power rule

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

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

16 a^{27} b^{2 + 12} (3^{2} {(a^{9})}^{2})

$16 a^{27} b^{2 + 12} (3^{2} {(a^{9})}^{2})$

Step 7.3

Add

2

$2$ and

12

$12$ .

16 a^{27} b^{14} (3^{2} {(a^{9})}^{2})

$16 a^{27} b^{14} (3^{2} {(a^{9})}^{2})$

16 a^{27} b^{14} (3^{2} {(a^{9})}^{2})

$16 a^{27} b^{14} (3^{2} {(a^{9})}^{2})$

Step 8

Raise

3

$3$ to the power of

2

$2$ .

16 a^{27} b^{14} (9 {(a^{9})}^{2})

$16 a^{27} b^{14} (9 {(a^{9})}^{2})$

Step 9

Multiply the exponents in

{(a^{9})}^{2}

${(a^{9})}^{2}$ .

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Step 9.1

Apply the power rule and multiply exponents,

{(a^{m})}^{n} = a^{m n}

${(a^{m})}^{n} = a^{m n}$ .

16 a^{27} b^{14} (9 a^{9 \cdot 2})

$16 a^{27} b^{14} (9 a^{9 \cdot 2})$

Step 9.2

Multiply

9

$9$ by

$2$ .

$16 a^{27} b^{14} (9 a^{18})$

Step 10

Multiply

$a^{27}$ by

$a^{18}$ by adding the exponents.

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Step 10.1

Move

$a^{18}$ .

$16 (a^{18} a^{27}) b^{14} \cdot 9$

Step 10.2

Use the power rule

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

$16 a^{18 + 27} b^{14} \cdot 9$

Step 10.3

Add

$18$ and

$27$ .

$16 a^{45} b^{14} \cdot 9$

Step 11

Multiply

$9$ by

$16$ .

$144 a^{45} b^{14}$