Basic Math Examples

$m \sqrt{18 m^{10}} \cdot (2 k \sqrt{200 k^{12}})$

Step 1

Rewrite using the commutative property of multiplication.

$2 m \sqrt{18 m^{10}} k \sqrt{200 k^{12}}$

Step 2

Rewrite $18 m^{10}$ as ${(3 m^{5})}^{2} \cdot 2$ .

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Step 2.1

Factor $9$ out of $18$ .

$2 m \sqrt{9 (2) m^{10}} k \sqrt{200 k^{12}}$

Step 2.2

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

$2 m \sqrt{3^{2} \cdot 2 m^{10}} k \sqrt{200 k^{12}}$

Step 2.3

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

$2 m \sqrt{3^{2} \cdot 2 {(m^{5})}^{2}} k \sqrt{200 k^{12}}$

Step 2.4

Move $2$ .

$2 m \sqrt{3^{2} {(m^{5})}^{2} \cdot 2} k \sqrt{200 k^{12}}$

Step 2.5

Rewrite $3^{2} {(m^{5})}^{2}$ as ${(3 m^{5})}^{2}$ .

$2 m \sqrt{{(3 m^{5})}^{2} \cdot 2} k \sqrt{200 k^{12}}$

Step 3

Pull terms out from under the radical.

$2 m (3 m^{5} \sqrt{2}) k \sqrt{200 k^{12}}$

Step 4

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

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Step 4.1

Move $m^{5}$ .

$2 (m^{5} m) (3 \sqrt{2}) k \sqrt{200 k^{12}}$

Step 4.2

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

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Step 4.2.1

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

$2 (m^{5} m^{1}) (3 \sqrt{2}) k \sqrt{200 k^{12}}$

Step 4.2.2

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

$2 m^{5 + 1} (3 \sqrt{2}) k \sqrt{200 k^{12}}$

Step 4.3

Add $5$ and $1$ .

$2 m^{6} (3 \sqrt{2}) k \sqrt{200 k^{12}}$

Step 5

Multiply $3$ by $2$ .

$6 m^{6} \sqrt{2} k \sqrt{200 k^{12}}$

Step 6

Rewrite $200 k^{12}$ as ${(10 k^{6})}^{2} \cdot 2$ .

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Step 6.1

Factor $100$ out of $200$ .

$6 m^{6} \sqrt{2} k \sqrt{100 (2) k^{12}}$

Step 6.2

Rewrite $100$ as $10^{2}$ .

$6 m^{6} \sqrt{2} k \sqrt{10^{2} \cdot 2 k^{12}}$

Step 6.3

Rewrite $k^{12}$ as ${(k^{6})}^{2}$ .

$6 m^{6} \sqrt{2} k \sqrt{10^{2} \cdot 2 {(k^{6})}^{2}}$

Step 6.4

Move $2$ .

$6 m^{6} \sqrt{2} k \sqrt{10^{2} {(k^{6})}^{2} \cdot 2}$

Step 6.5

Rewrite $10^{2} {(k^{6})}^{2}$ as ${(10 k^{6})}^{2}$ .

$6 m^{6} \sqrt{2} k \sqrt{{(10 k^{6})}^{2} \cdot 2}$

Step 7

Pull terms out from under the radical.

$6 m^{6} \sqrt{2} k (10 k^{6} \sqrt{2})$

Step 8

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

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Step 8.1

Move $k^{6}$ .

$6 m^{6} \sqrt{2} (k^{6} k) (10 \sqrt{2})$

Step 8.2

Multiply $k^{6}$ by $k$ .

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Step 8.2.1

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

$6 m^{6} \sqrt{2} (k^{6} k^{1}) (10 \sqrt{2})$

Step 8.2.2

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

$6 m^{6} \sqrt{2} k^{6 + 1} (10 \sqrt{2})$

Step 8.3

Add $6$ and $1$ .

$6 m^{6} \sqrt{2} k^{7} (10 \sqrt{2})$

Step 9

Multiply $6 m^{6} \sqrt{2} k^{7} (10 \sqrt{2})$ .

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Step 9.1

Multiply $10$ by $6$ .

$60 m^{6} \sqrt{2} k^{7} \sqrt{2}$

Step 9.2

Raise $\sqrt{2}$ to the power of $1$ .

$60 m^{6} k^{7} ({\sqrt{2}}^{1} \sqrt{2})$

Step 9.3

Raise $\sqrt{2}$ to the power of $1$ .

$60 m^{6} k^{7} ({\sqrt{2}}^{1} {\sqrt{2}}^{1})$

Step 9.4

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

$60 m^{6} k^{7} {\sqrt{2}}^{1 + 1}$

Step 9.5

Add $1$ and $1$ .

$60 m^{6} k^{7} {\sqrt{2}}^{2}$

Step 10

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

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Step 10.1

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

$60 m^{6} k^{7} {(2^{\frac{1}{2}})}^{2}$

Step 10.2

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

$60 m^{6} k^{7} \cdot 2^{\frac{1}{2} \cdot 2}$

Step 10.3

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

$60 m^{6} k^{7} \cdot 2^{\frac{2}{2}}$

Step 10.4

Cancel the common factor of $2$ .

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Step 10.4.1

Cancel the common factor.

$60 m^{6} k^{7} \cdot 2^{\frac{2}{2}}$

Step 10.4.2

Rewrite the expression.

$60 m^{6} k^{7} \cdot 2^{1}$

Step 10.5

Evaluate the exponent.

$60 m^{6} k^{7} \cdot 2$

Step 11

Multiply $2$ by $60$ .

$120 m^{6} k^{7}$