Basic Math Examples

${(2 c^{3} d^{2})}^{5} {(\frac{c^{6} d^{8}}{4 c^{2} d})}^{3}$

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 $2 c^{3} d^{2}$ .

${(2 c^{3})}^{5} {(d^{2})}^{5} {(\frac{c^{6} d^{8}}{4 c^{2} d})}^{3}$

Step 1.2

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

$2^{5} {(c^{3})}^{5} {(d^{2})}^{5} {(\frac{c^{6} d^{8}}{4 c^{2} d})}^{3}$

Step 2

Simplify the expression.

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Step 2.1

Raise $2$ to the power of $5$ .

$32 {(c^{3})}^{5} {(d^{2})}^{5} {(\frac{c^{6} d^{8}}{4 c^{2} d})}^{3}$

Step 2.2

Multiply the exponents in ${(c^{3})}^{5}$ .

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Step 2.2.1

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

$32 c^{3 \cdot 5} {(d^{2})}^{5} {(\frac{c^{6} d^{8}}{4 c^{2} d})}^{3}$

Step 2.2.2

Multiply $3$ by $5$ .

$32 c^{15} {(d^{2})}^{5} {(\frac{c^{6} d^{8}}{4 c^{2} d})}^{3}$

Step 2.3

Multiply the exponents in ${(d^{2})}^{5}$ .

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Step 2.3.1

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

$32 c^{15} d^{2 \cdot 5} {(\frac{c^{6} d^{8}}{4 c^{2} d})}^{3}$

Step 2.3.2

Multiply $2$ by $5$ .

$32 c^{15} d^{10} {(\frac{c^{6} d^{8}}{4 c^{2} d})}^{3}$

Step 3

Cancel the common factor of $c^{6}$ and $c^{2}$ .

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Step 3.1

Factor $c^{2}$ out of $c^{6} d^{8}$ .

$32 c^{15} d^{10} {(\frac{c^{2} (c^{4} d^{8})}{4 c^{2} d})}^{3}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $c^{2}$ out of $4 c^{2} d$ .

$32 c^{15} d^{10} {(\frac{c^{2} (c^{4} d^{8})}{c^{2} (4 d)})}^{3}$

Step 3.2.2

Cancel the common factor.

$32 c^{15} d^{10} {(\frac{c^{2} (c^{4} d^{8})}{c^{2} (4 d)})}^{3}$

Step 3.2.3

Rewrite the expression.

$32 c^{15} d^{10} {(\frac{c^{4} d^{8}}{4 d})}^{3}$

Step 4

Cancel the common factor of $d^{8}$ and $d$ .

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Step 4.1

Factor $d$ out of $c^{4} d^{8}$ .

$32 c^{15} d^{10} {(\frac{d (c^{4} d^{7})}{4 d})}^{3}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $d$ out of $4 d$ .

$32 c^{15} d^{10} {(\frac{d (c^{4} d^{7})}{d \cdot 4})}^{3}$

Step 4.2.2

Cancel the common factor.

$32 c^{15} d^{10} {(\frac{d (c^{4} d^{7})}{d \cdot 4})}^{3}$

Step 4.2.3

Rewrite the expression.

$32 c^{15} d^{10} {(\frac{c^{4} d^{7}}{4})}^{3}$

Step 5

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

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Step 5.1

Apply the product rule to $\frac{c^{4} d^{7}}{4}$ .

$32 c^{15} d^{10} \frac{{(c^{4} d^{7})}^{3}}{4^{3}}$

Step 5.2

Apply the product rule to $c^{4} d^{7}$ .

$32 c^{15} d^{10} \frac{{(c^{4})}^{3} {(d^{7})}^{3}}{4^{3}}$

Step 6

Simplify the numerator.

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Step 6.1

Multiply the exponents in ${(c^{4})}^{3}$ .

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Step 6.1.1

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

$32 c^{15} d^{10} \frac{c^{4 \cdot 3} {(d^{7})}^{3}}{4^{3}}$

Step 6.1.2

Multiply $4$ by $3$ .

$32 c^{15} d^{10} \frac{c^{12} {(d^{7})}^{3}}{4^{3}}$

Step 6.2

Multiply the exponents in ${(d^{7})}^{3}$ .

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Step 6.2.1

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

$32 c^{15} d^{10} \frac{c^{12} d^{7 \cdot 3}}{4^{3}}$

Step 6.2.2

Multiply $7$ by $3$ .

$32 c^{15} d^{10} \frac{c^{12} d^{21}}{4^{3}}$

Step 7

Simplify terms.

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Step 7.1

Raise $4$ to the power of $3$ .

$32 c^{15} d^{10} \frac{c^{12} d^{21}}{64}$

Step 7.2

Cancel the common factor of $32$ .

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Step 7.2.1

Factor $32$ out of $32 c^{15} d^{10}$ .

$32 (c^{15} d^{10}) \frac{c^{12} d^{21}}{64}$

Step 7.2.2

Factor $32$ out of $64$ .

$32 (c^{15} d^{10}) \frac{c^{12} d^{21}}{32 (2)}$

Step 7.2.3

Cancel the common factor.

$32 (c^{15} d^{10}) \frac{c^{12} d^{21}}{32 \cdot 2}$

Step 7.2.4

Rewrite the expression.

$c^{15} d^{10} \frac{c^{12} d^{21}}{2}$

Step 7.3

Combine $c^{15}$ and $\frac{c^{12} d^{21}}{2}$ .

$d^{10} \frac{c^{15} (c^{12} d^{21})}{2}$

Step 8

Multiply $c^{15}$ by $c^{12}$ by adding the exponents.

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Step 8.1

Move $c^{12}$ .

$d^{10} \frac{c^{12} c^{15} d^{21}}{2}$

Step 8.2

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

$d^{10} \frac{c^{12 + 15} d^{21}}{2}$

Step 8.3

Add $12$ and $15$ .

$d^{10} \frac{c^{27} d^{21}}{2}$

Step 9

Combine $d^{10}$ and $\frac{c^{27} d^{21}}{2}$ .

$\frac{d^{10} (c^{27} d^{21})}{2}$

Step 10

Multiply $d^{10}$ by $d^{21}$ by adding the exponents.

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Step 10.1

Move $d^{21}$ .

$\frac{d^{21} d^{10} c^{27}}{2}$

Step 10.2

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

$\frac{d^{21 + 10} c^{27}}{2}$

Step 10.3

Add $21$ and $10$ .

$\frac{d^{31} c^{27}}{2}$