Basic Math Examples

$- \frac{1}{2} \cdot \sqrt{27 c^{2}} \cdot (- 6 \sqrt{12 c^{10}})$

Step 1

Rewrite $27 c^{2}$ as ${(3 c)}^{2} \cdot 3$ .

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Step 1.1

Factor $9$ out of $27$ .

$- \frac{1}{2} \cdot \sqrt{9 (3) c^{2}} \cdot (- 6 \sqrt{12 c^{10}})$

Step 1.2

Rewrite $9$ as $3^{2}$ .

$- \frac{1}{2} \cdot \sqrt{3^{2} \cdot 3 c^{2}} \cdot (- 6 \sqrt{12 c^{10}})$

Step 1.3

Move $3$ .

$- \frac{1}{2} \cdot \sqrt{3^{2} c^{2} \cdot 3} \cdot (- 6 \sqrt{12 c^{10}})$

Step 1.4

Rewrite $3^{2} c^{2}$ as ${(3 c)}^{2}$ .

$- \frac{1}{2} \cdot \sqrt{{(3 c)}^{2} \cdot 3} \cdot (- 6 \sqrt{12 c^{10}})$

Step 2

Pull terms out from under the radical.

$- \frac{1}{2} \cdot (3 c \sqrt{3}) \cdot (- 6 \sqrt{12 c^{10}})$

Step 3

Multiply $- \frac{1}{2} (3 c \sqrt{3})$ .

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Step 3.1

Multiply $3$ by $- 1$ .

$- 3 (\frac{1}{2}) (c \sqrt{3}) \cdot (- 6 \sqrt{12 c^{10}})$

Step 3.2

Combine $- 3$ and $\frac{1}{2}$ .

$\frac{- 3}{2} (c \sqrt{3}) \cdot (- 6 \sqrt{12 c^{10}})$

Step 3.3

Combine $c$ and $\frac{- 3}{2}$ .

$\frac{c \cdot - 3}{2} \sqrt{3} \cdot (- 6 \sqrt{12 c^{10}})$

Step 3.4

Combine $\frac{c \cdot - 3}{2}$ and $\sqrt{3}$ .

$\frac{c \cdot - 3 \sqrt{3}}{2} \cdot (- 6 \sqrt{12 c^{10}})$

Step 4

Rewrite using the commutative property of multiplication.

$- 6 \frac{c \cdot - 3 \sqrt{3}}{2} \sqrt{12 c^{10}}$

Step 5

Cancel the common factor of $2$ .

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Step 5.1

Factor $2$ out of $- 6$ .

$2 (- 3) \frac{c \cdot - 3 \sqrt{3}}{2} \sqrt{12 c^{10}}$

Step 5.2

Cancel the common factor.

$2 \cdot - 3 \frac{c \cdot - 3 \sqrt{3}}{2} \sqrt{12 c^{10}}$

Step 5.3

Rewrite the expression.

$- 3 (c \cdot - 3 \sqrt{3}) \sqrt{12 c^{10}}$

Step 6

Multiply $- 3$ by $- 3$ .

$9 (c \sqrt{3}) \sqrt{12 c^{10}}$

Step 7

Rewrite $12 c^{10}$ as ${(2 c^{5})}^{2} \cdot 3$ .

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Step 7.1

Factor $4$ out of $12$ .

$9 c \sqrt{3} \sqrt{4 (3) c^{10}}$

Step 7.2

Rewrite $4$ as $2^{2}$ .

$9 c \sqrt{3} \sqrt{2^{2} \cdot 3 c^{10}}$

Step 7.3

Rewrite $c^{10}$ as ${(c^{5})}^{2}$ .

$9 c \sqrt{3} \sqrt{2^{2} \cdot 3 {(c^{5})}^{2}}$

Step 7.4

Move $3$ .

$9 c \sqrt{3} \sqrt{2^{2} {(c^{5})}^{2} \cdot 3}$

Step 7.5

Rewrite $2^{2} {(c^{5})}^{2}$ as ${(2 c^{5})}^{2}$ .

$9 c \sqrt{3} \sqrt{{(2 c^{5})}^{2} \cdot 3}$

Step 8

Pull terms out from under the radical.

$9 c \sqrt{3} (2 c^{5} \sqrt{3})$

Step 9

Multiply $c$ by $c^{5}$ by adding the exponents.

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Step 9.1

Move $c^{5}$ .

$9 (c^{5} c) \sqrt{3} (2 \sqrt{3})$

Step 9.2

Multiply $c^{5}$ by $c$ .

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Step 9.2.1

Raise $c$ to the power of $1$ .

$9 (c^{5} c^{1}) \sqrt{3} (2 \sqrt{3})$

Step 9.2.2

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

$9 c^{5 + 1} \sqrt{3} (2 \sqrt{3})$

Step 9.3

Add $5$ and $1$ .

$9 c^{6} \sqrt{3} (2 \sqrt{3})$

Step 10

Multiply $9 c^{6} \sqrt{3} (2 \sqrt{3})$ .

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Step 10.1

Multiply $2$ by $9$ .

$18 c^{6} \sqrt{3} \sqrt{3}$

Step 10.2

Raise $\sqrt{3}$ to the power of $1$ .

$18 c^{6} ({\sqrt{3}}^{1} \sqrt{3})$

Step 10.3

Raise $\sqrt{3}$ to the power of $1$ .

$18 c^{6} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 10.4

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

$18 c^{6} {\sqrt{3}}^{1 + 1}$

Step 10.5

Add $1$ and $1$ .

$18 c^{6} {\sqrt{3}}^{2}$

Step 11

Rewrite ${\sqrt{3}}^{2}$ as $3$ .

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Step 11.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3}$ as $3^{\frac{1}{2}}$ .

$18 c^{6} {(3^{\frac{1}{2}})}^{2}$

Step 11.2

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

$18 c^{6} \cdot 3^{\frac{1}{2} \cdot 2}$

Step 11.3

Combine $\frac{1}{2}$ and $2$ .

$18 c^{6} \cdot 3^{\frac{2}{2}}$

Step 11.4

Cancel the common factor of $2$ .

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Step 11.4.1

Cancel the common factor.

$18 c^{6} \cdot 3^{\frac{2}{2}}$

Step 11.4.2

Rewrite the expression.

$18 c^{6} \cdot 3^{1}$

Step 11.5

Evaluate the exponent.

$18 c^{6} \cdot 3$

Step 12

Multiply $3$ by $18$ .

$54 c^{6}$