Calculus Examples

$\sqrt{14 m^{5}} \cdot \sqrt{7 m^{13}}$

Step 1

Rewrite $14 m^{5}$ as ${(m^{2})}^{2} \cdot (14 m)$ .

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Step 1.1

Factor out $m^{4}$ .

$\sqrt{14 (m^{4} m)} \cdot \sqrt{7 m^{13}}$

Step 1.2

Rewrite $m^{4}$ as ${(m^{2})}^{2}$ .

$\sqrt{14 ({(m^{2})}^{2} m)} \cdot \sqrt{7 m^{13}}$

Step 1.3

Reorder $14$ and ${(m^{2})}^{2}$ .

$\sqrt{{(m^{2})}^{2} \cdot 14 m} \cdot \sqrt{7 m^{13}}$

Step 1.4

Add parentheses.

$\sqrt{{(m^{2})}^{2} \cdot (14 m)} \cdot \sqrt{7 m^{13}}$

Step 2

Pull terms out from under the radical.

$m^{2} \sqrt{14 m} \cdot \sqrt{7 m^{13}}$

Step 3

Rewrite $7 m^{13}$ as ${({(m^{3})}^{2})}^{2} \cdot (7 m)$ .

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Step 3.1

Factor out $m^{12}$ .

$m^{2} \sqrt{14 m} \cdot \sqrt{7 (m^{12} m)}$

Step 3.2

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

$m^{2} \sqrt{14 m} \cdot \sqrt{7 ({(m^{6})}^{2} m)}$

Step 3.3

Reorder $7$ and ${(m^{6})}^{2}$ .

$m^{2} \sqrt{14 m} \cdot \sqrt{{(m^{6})}^{2} \cdot 7 m}$

Step 3.4

Rewrite $m^{6}$ as ${(m^{3})}^{2}$ .

$m^{2} \sqrt{14 m} \cdot \sqrt{{({(m^{3})}^{2})}^{2} \cdot 7 m}$

Step 3.5

Add parentheses.

$m^{2} \sqrt{14 m} \cdot \sqrt{{({(m^{3})}^{2})}^{2} \cdot (7 m)}$

Step 4

Pull terms out from under the radical.

$m^{2} \sqrt{14 m} \cdot ({(m^{3})}^{2} \sqrt{7 m})$

Step 5

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

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Step 5.1

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

$m^{2} \sqrt{14 m} \cdot (m^{3 \cdot 2} \sqrt{7 m})$

Step 5.2

Multiply $3$ by $2$ .

$m^{2} \sqrt{14 m} \cdot (m^{6} \sqrt{7 m})$

Step 6

Multiply $m^{2}$ by $m^{6}$ by adding the exponents.

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Step 6.1

Move $m^{6}$ .

$m^{6} m^{2} \sqrt{14 m} \cdot \sqrt{7 m}$

Step 6.2

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

$m^{6 + 2} \sqrt{14 m} \cdot \sqrt{7 m}$

Step 6.3

Add $6$ and $2$ .

$m^{8} \sqrt{14 m} \cdot \sqrt{7 m}$

Step 7

Multiply $m^{8} \sqrt{14 m} \sqrt{7 m}$ .

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Step 7.1

Combine using the product rule for radicals.

$m^{8} \sqrt{7 m (14 m)}$

Step 7.2

Multiply $14$ by $7$ .

$m^{8} \sqrt{98 m \cdot m}$

Step 7.3

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

$m^{8} \sqrt{98 (m^{1} m)}$

Step 7.4

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

$m^{8} \sqrt{98 (m^{1} m^{1})}$

Step 7.5

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

$m^{8} \sqrt{98 m^{1 + 1}}$

Step 7.6

Add $1$ and $1$ .

$m^{8} \sqrt{98 m^{2}}$

Step 8

Rewrite $98 m^{2}$ as ${(7 m)}^{2} \cdot 2$ .

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Step 8.1

Factor $49$ out of $98$ .

$m^{8} \sqrt{49 (2) m^{2}}$

Step 8.2

Rewrite $49$ as $7^{2}$ .

$m^{8} \sqrt{7^{2} \cdot 2 m^{2}}$

Step 8.3

Move $2$ .

$m^{8} \sqrt{7^{2} m^{2} \cdot 2}$

Step 8.4

Rewrite $7^{2} m^{2}$ as ${(7 m)}^{2}$ .

$m^{8} \sqrt{{(7 m)}^{2} \cdot 2}$

Step 9

Pull terms out from under the radical.

$m^{8} (7 m \sqrt{2})$

Step 10

Rewrite using the commutative property of multiplication.

$7 \sqrt{2} m^{8} m$

Step 11

Multiply $m^{8}$ by $m$ by adding the exponents.

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Step 11.1

Move $m$ .

$7 \sqrt{2} (m \cdot m^{8})$

Step 11.2

Multiply $m$ by $m^{8}$ .

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Step 11.2.1

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

$7 \sqrt{2} (m^{1} m^{8})$

Step 11.2.2

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

$7 \sqrt{2} m^{1 + 8}$

Step 11.3

Add $1$ and $8$ .

$7 \sqrt{2} m^{9}$