Basic Math Examples

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

Step 1

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

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Step 1.1

Apply the product rule to $3 a^{2} b^{3} c^{4}$ .

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

Step 1.2

Apply the product rule to $3 a^{2} b^{3}$ .

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

Step 1.3

Apply the product rule to $3 a^{2}$ .

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

Step 2

Raise $3$ to the power of $2$ .

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

Step 3

Multiply the exponents in ${(a^{2})}^{2}$ .

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Step 3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 3.2

Multiply $2$ by $2$ .

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

Step 4

Multiply the exponents in ${(b^{3})}^{2}$ .

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Step 4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 4.2

Multiply $3$ by $2$ .

$9 a^{4} b^{6} {(c^{4})}^{2} \cdot {(2 a b^{2} c^{3})}^{3}$

Step 5

Multiply the exponents in ${(c^{4})}^{2}$ .

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Step 5.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$9 a^{4} b^{6} c^{4 \cdot 2} \cdot {(2 a b^{2} c^{3})}^{3}$

Step 5.2

Multiply $4$ by $2$ .

$9 a^{4} b^{6} c^{8} \cdot {(2 a b^{2} c^{3})}^{3}$

Step 6

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

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Step 6.1

Apply the product rule to $2 a b^{2} c^{3}$ .

$9 a^{4} b^{6} c^{8} \cdot ({(2 a b^{2})}^{3} {(c^{3})}^{3})$

Step 6.2

Apply the product rule to $2 a b^{2}$ .

$9 a^{4} b^{6} c^{8} \cdot ({(2 a)}^{3} {(b^{2})}^{3} {(c^{3})}^{3})$

Step 6.3

Apply the product rule to $2 a$ .

$9 a^{4} b^{6} c^{8} \cdot (2^{3} a^{3} {(b^{2})}^{3} {(c^{3})}^{3})$

Step 7

Multiply $a^{4}$ by $a^{3}$ by adding the exponents.

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Step 7.1

Move $a^{3}$ .

$9 (a^{3} a^{4}) b^{6} c^{8} \cdot (2^{3} {(b^{2})}^{3} {(c^{3})}^{3})$

Step 7.2

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

$9 a^{3 + 4} b^{6} c^{8} \cdot (2^{3} {(b^{2})}^{3} {(c^{3})}^{3})$

Step 7.3

Add $3$ and $4$ .

$9 a^{7} b^{6} c^{8} \cdot (2^{3} {(b^{2})}^{3} {(c^{3})}^{3})$

Step 8

Raise $2$ to the power of $3$ .

$9 a^{7} b^{6} c^{8} \cdot (8 {(b^{2})}^{3} {(c^{3})}^{3})$

Step 9

Multiply the exponents in ${(b^{2})}^{3}$ .

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Step 9.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$9 a^{7} b^{6} c^{8} \cdot (8 b^{2 \cdot 3} {(c^{3})}^{3})$

Step 9.2

Multiply $2$ by $3$ .

$9 a^{7} b^{6} c^{8} \cdot (8 b^{6} {(c^{3})}^{3})$

Step 10

Multiply $b^{6}$ by $b^{6}$ by adding the exponents.

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Step 10.1

Move $b^{6}$ .

$9 a^{7} (b^{6} b^{6}) c^{8} \cdot (8 {(c^{3})}^{3})$

Step 10.2

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

$9 a^{7} b^{6 + 6} c^{8} \cdot (8 {(c^{3})}^{3})$

Step 10.3

Add $6$ and $6$ .

$9 a^{7} b^{12} c^{8} \cdot (8 {(c^{3})}^{3})$

Step 11

Multiply the exponents in ${(c^{3})}^{3}$ .

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Step 11.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$9 a^{7} b^{12} c^{8} \cdot (8 c^{3 \cdot 3})$

Step 11.2

Multiply $3$ by $3$ .

$9 a^{7} b^{12} c^{8} \cdot (8 c^{9})$

Step 12

Multiply $c^{8}$ by $c^{9}$ by adding the exponents.

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Step 12.1

Move $c^{9}$ .

$9 a^{7} b^{12} (c^{9} c^{8}) \cdot 8$

Step 12.2

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

$9 a^{7} b^{12} c^{9 + 8} \cdot 8$

Step 12.3

Add $9$ and $8$ .

$9 a^{7} b^{12} c^{17} \cdot 8$

Step 13

Multiply $8$ by $9$ .

$72 a^{7} b^{12} c^{17}$