Basic Math Examples

$(\sqrt{3 m n^{3}} - \sqrt{5 n^{7}}) (\sqrt{3 m} + \sqrt{5 n^{7}})$

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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Step 1.1.1.1

Factor out $n^{2}$ .

$(\sqrt{3 m (n^{2} n)} - \sqrt{5 n^{7}}) (\sqrt{3 m} + \sqrt{5 n^{7}})$

Step 1.1.1.2

Move $m$ .

$(\sqrt{3 n^{2} m n} - \sqrt{5 n^{7}}) (\sqrt{3 m} + \sqrt{5 n^{7}})$

Step 1.1.1.3

Reorder $3$ and $n^{2}$ .

$(\sqrt{n^{2} \cdot 3 m n} - \sqrt{5 n^{7}}) (\sqrt{3 m} + \sqrt{5 n^{7}})$

Step 1.1.1.4

Add parentheses.

$(\sqrt{n^{2} \cdot 3 (m n)} - \sqrt{5 n^{7}}) (\sqrt{3 m} + \sqrt{5 n^{7}})$

Step 1.1.1.5

Add parentheses.

$(\sqrt{n^{2} \cdot (3 m n)} - \sqrt{5 n^{7}}) (\sqrt{3 m} + \sqrt{5 n^{7}})$

Step 1.1.2

Pull terms out from under the radical.

$(n \sqrt{3 m n} - \sqrt{5 n^{7}}) (\sqrt{3 m} + \sqrt{5 n^{7}})$

Step 1.1.3

Rewrite $5 n^{7}$ as ${(n^{3})}^{2} \cdot (5 n)$ .

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Step 1.1.3.1

Factor out $n^{6}$ .

$(n \sqrt{3 m n} - \sqrt{5 (n^{6} n)}) (\sqrt{3 m} + \sqrt{5 n^{7}})$

Step 1.1.3.2

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

$(n \sqrt{3 m n} - \sqrt{5 ({(n^{3})}^{2} n)}) (\sqrt{3 m} + \sqrt{5 n^{7}})$

Step 1.1.3.3

Reorder $5$ and ${(n^{3})}^{2}$ .

$(n \sqrt{3 m n} - \sqrt{{(n^{3})}^{2} \cdot 5 n}) (\sqrt{3 m} + \sqrt{5 n^{7}})$

Step 1.1.3.4

Add parentheses.

$(n \sqrt{3 m n} - \sqrt{{(n^{3})}^{2} \cdot (5 n)}) (\sqrt{3 m} + \sqrt{5 n^{7}})$

Step 1.1.4

Pull terms out from under the radical.

$(n \sqrt{3 m n} - n^{3} \sqrt{5 n}) (\sqrt{3 m} + \sqrt{5 n^{7}})$

Step 1.2

Simplify each term.

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Step 1.2.1

Rewrite $5 n^{7}$ as ${(n^{3})}^{2} \cdot (5 n)$ .

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Step 1.2.1.1

Factor out $n^{6}$ .

$(n \sqrt{3 m n} - n^{3} \sqrt{5 n}) (\sqrt{3 m} + \sqrt{5 (n^{6} n)})$

Step 1.2.1.2

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

$(n \sqrt{3 m n} - n^{3} \sqrt{5 n}) (\sqrt{3 m} + \sqrt{5 ({(n^{3})}^{2} n)})$

Step 1.2.1.3

Reorder $5$ and ${(n^{3})}^{2}$ .

$(n \sqrt{3 m n} - n^{3} \sqrt{5 n}) (\sqrt{3 m} + \sqrt{{(n^{3})}^{2} \cdot 5 n})$

Step 1.2.1.4

Add parentheses.

$(n \sqrt{3 m n} - n^{3} \sqrt{5 n}) (\sqrt{3 m} + \sqrt{{(n^{3})}^{2} \cdot (5 n)})$

Step 1.2.2

Pull terms out from under the radical.

$(n \sqrt{3 m n} - n^{3} \sqrt{5 n}) (\sqrt{3 m} + n^{3} \sqrt{5 n})$

Step 2

Expand $(n \sqrt{3 m n} - n^{3} \sqrt{5 n}) (\sqrt{3 m} + n^{3} \sqrt{5 n})$ using the FOIL Method.

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Step 2.1

Apply the distributive property.

$n \sqrt{3 m n} (\sqrt{3 m} + n^{3} \sqrt{5 n}) - n^{3} \sqrt{5 n} (\sqrt{3 m} + n^{3} \sqrt{5 n})$

Step 2.2

Apply the distributive property.

$n \sqrt{3 m n} \sqrt{3 m} + n \sqrt{3 m n} (n^{3} \sqrt{5 n}) - n^{3} \sqrt{5 n} (\sqrt{3 m} + n^{3} \sqrt{5 n})$

Step 2.3

Apply the distributive property.

$n \sqrt{3 m n} \sqrt{3 m} + n \sqrt{3 m n} (n^{3} \sqrt{5 n}) - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3

Simplify terms.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply $n \sqrt{3 m n} \sqrt{3 m}$ .

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Step 3.1.1.1

Combine using the product rule for radicals.

$n \sqrt{3 m (3 m n)} + n \sqrt{3 m n} (n^{3} \sqrt{5 n}) - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.1.2

Multiply $3$ by $3$ .

$n \sqrt{9 m (m n)} + n \sqrt{3 m n} (n^{3} \sqrt{5 n}) - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.1.3

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

$n \sqrt{9 (m^{1} m) n} + n \sqrt{3 m n} (n^{3} \sqrt{5 n}) - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.1.4

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

$n \sqrt{9 (m^{1} m^{1}) n} + n \sqrt{3 m n} (n^{3} \sqrt{5 n}) - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.1.5

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

$n \sqrt{9 m^{1 + 1} n} + n \sqrt{3 m n} (n^{3} \sqrt{5 n}) - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.1.6

Add $1$ and $1$ .

$n \sqrt{9 m^{2} n} + n \sqrt{3 m n} (n^{3} \sqrt{5 n}) - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.2

Rewrite $9 m^{2} n$ as ${(3 m)}^{2} n$ .

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Step 3.1.2.1

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

$n \sqrt{3^{2} m^{2} n} + n \sqrt{3 m n} (n^{3} \sqrt{5 n}) - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.2.2

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

$n \sqrt{{(3 m)}^{2} n} + n \sqrt{3 m n} (n^{3} \sqrt{5 n}) - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.3

Pull terms out from under the radical.

$n (3 m \sqrt{n}) + n \sqrt{3 m n} (n^{3} \sqrt{5 n}) - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.4

Rewrite using the commutative property of multiplication.

$3 n (m \sqrt{n}) + n \sqrt{3 m n} (n^{3} \sqrt{5 n}) - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.5

Multiply $n$ by $n^{3}$ by adding the exponents.

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Step 3.1.5.1

Move $n^{3}$ .

$3 n (m \sqrt{n}) + n^{3} n \sqrt{3 m n} \sqrt{5 n} - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.5.2

Multiply $n^{3}$ by $n$ .

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Step 3.1.5.2.1

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

$3 n (m \sqrt{n}) + n^{3} n^{1} \sqrt{3 m n} \sqrt{5 n} - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.5.2.2

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

$3 n (m \sqrt{n}) + n^{3 + 1} \sqrt{3 m n} \sqrt{5 n} - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.5.3

Add $3$ and $1$ .

$3 n (m \sqrt{n}) + n^{4} \sqrt{3 m n} \sqrt{5 n} - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.6

Multiply $n^{4} \sqrt{3 m n} \sqrt{5 n}$ .

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Step 3.1.6.1

Combine using the product rule for radicals.

$3 n (m \sqrt{n}) + n^{4} \sqrt{5 n (3 m n)} - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.6.2

Multiply $3$ by $5$ .

$3 n (m \sqrt{n}) + n^{4} \sqrt{15 n (m n)} - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.6.3

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

$3 n (m \sqrt{n}) + n^{4} \sqrt{15 (m (n^{1} n))} - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.6.4

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

$3 n (m \sqrt{n}) + n^{4} \sqrt{15 (m (n^{1} n^{1}))} - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.6.5

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

$3 n (m \sqrt{n}) + n^{4} \sqrt{15 (m n^{1 + 1})} - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.6.6

Add $1$ and $1$ .

$3 n (m \sqrt{n}) + n^{4} \sqrt{15 (m n^{2})} - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.7

Rewrite $15 m n^{2}$ as $n^{2} \cdot (15 m)$ .

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Step 3.1.7.1

Move $m$ .

$3 n (m \sqrt{n}) + n^{4} \sqrt{15 n^{2} m} - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.7.2

Reorder $15$ and $n^{2}$ .

$3 n (m \sqrt{n}) + n^{4} \sqrt{n^{2} \cdot 15 m} - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.7.3

Add parentheses.

$3 n (m \sqrt{n}) + n^{4} \sqrt{n^{2} \cdot (15 m)} - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.8

Pull terms out from under the radical.

$3 n (m \sqrt{n}) + n^{4} (n \sqrt{15 m}) - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.9

Multiply $n^{4}$ by $n$ by adding the exponents.

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Step 3.1.9.1

Move $n$ .

$3 n (m \sqrt{n}) + n \cdot n^{4} \sqrt{15 m} - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.9.2

Multiply $n$ by $n^{4}$ .

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Step 3.1.9.2.1

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

$3 n (m \sqrt{n}) + n^{1} n^{4} \sqrt{15 m} - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.9.2.2

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

$3 n (m \sqrt{n}) + n^{1 + 4} \sqrt{15 m} - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.9.3

Add $1$ and $4$ .

$3 n (m \sqrt{n}) + n^{5} \sqrt{15 m} - n^{3} \sqrt{5 n} \sqrt{3 m} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.10

Multiply $- n^{3} \sqrt{5 n} \sqrt{3 m}$ .

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Step 3.1.10.1

Combine using the product rule for radicals.

$3 n (m \sqrt{n}) + n^{5} \sqrt{15 m} - n^{3} \sqrt{3 m (5 n)} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.10.2

Multiply $5$ by $3$ .

$3 n (m \sqrt{n}) + n^{5} \sqrt{15 m} - n^{3} \sqrt{15 m n} - n^{3} \sqrt{5 n} (n^{3} \sqrt{5 n})$

Step 3.1.11

Multiply $n^{3}$ by $n^{3}$ by adding the exponents.

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Step 3.1.11.1

Move $n^{3}$ .

$3 n (m \sqrt{n}) + n^{5} \sqrt{15 m} - n^{3} \sqrt{15 m n} - (n^{3} n^{3}) \sqrt{5 n} \sqrt{5 n}$

Step 3.1.11.2

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

$3 n (m \sqrt{n}) + n^{5} \sqrt{15 m} - n^{3} \sqrt{15 m n} - n^{3 + 3} \sqrt{5 n} \sqrt{5 n}$

Step 3.1.11.3

Add $3$ and $3$ .

$3 n (m \sqrt{n}) + n^{5} \sqrt{15 m} - n^{3} \sqrt{15 m n} - n^{6} \sqrt{5 n} \sqrt{5 n}$

Step 3.1.12

Multiply $- n^{6} \sqrt{5 n} \sqrt{5 n}$ .

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Step 3.1.12.1

Raise $\sqrt{5 n}$ to the power of $1$ .

$3 n (m \sqrt{n}) + n^{5} \sqrt{15 m} - n^{3} \sqrt{15 m n} - n^{6} ({\sqrt{5 n}}^{1} \sqrt{5 n})$

Step 3.1.12.2

Raise $\sqrt{5 n}$ to the power of $1$ .

$3 n (m \sqrt{n}) + n^{5} \sqrt{15 m} - n^{3} \sqrt{15 m n} - n^{6} ({\sqrt{5 n}}^{1} {\sqrt{5 n}}^{1})$

Step 3.1.12.3

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

$3 n (m \sqrt{n}) + n^{5} \sqrt{15 m} - n^{3} \sqrt{15 m n} - n^{6} {\sqrt{5 n}}^{1 + 1}$

Step 3.1.12.4

Add $1$ and $1$ .

$3 n (m \sqrt{n}) + n^{5} \sqrt{15 m} - n^{3} \sqrt{15 m n} - n^{6} {\sqrt{5 n}}^{2}$

Step 3.1.13

Rewrite ${\sqrt{5 n}}^{2}$ as $5 n$ .

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Step 3.1.13.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5 n}$ as ${(5 n)}^{\frac{1}{2}}$ .

$3 n (m \sqrt{n}) + n^{5} \sqrt{15 m} - n^{3} \sqrt{15 m n} - n^{6} {({(5 n)}^{\frac{1}{2}})}^{2}$

Step 3.1.13.2

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

$3 n (m \sqrt{n}) + n^{5} \sqrt{15 m} - n^{3} \sqrt{15 m n} - n^{6} {(5 n)}^{\frac{1}{2} \cdot 2}$

Step 3.1.13.3

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

$3 n (m \sqrt{n}) + n^{5} \sqrt{15 m} - n^{3} \sqrt{15 m n} - n^{6} {(5 n)}^{\frac{2}{2}}$

Step 3.1.13.4

Cancel the common factor of $2$ .

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Step 3.1.13.4.1

Cancel the common factor.

$3 n (m \sqrt{n}) + n^{5} \sqrt{15 m} - n^{3} \sqrt{15 m n} - n^{6} {(5 n)}^{\frac{2}{2}}$

Step 3.1.13.4.2

Rewrite the expression.

$3 n (m \sqrt{n}) + n^{5} \sqrt{15 m} - n^{3} \sqrt{15 m n} - n^{6} {(5 n)}^{1}$

Step 3.1.13.5

Simplify.

$3 n (m \sqrt{n}) + n^{5} \sqrt{15 m} - n^{3} \sqrt{15 m n} - n^{6} (5 n)$

Step 3.1.14

Rewrite using the commutative property of multiplication.

$3 n (m \sqrt{n}) + n^{5} \sqrt{15 m} - n^{3} \sqrt{15 m n} - 1 \cdot 5 n^{6} n$

Step 3.1.15

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

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Step 3.1.15.1

Move $n$ .

$3 n (m \sqrt{n}) + n^{5} \sqrt{15 m} - n^{3} \sqrt{15 m n} - 1 \cdot 5 (n \cdot n^{6})$

Step 3.1.15.2

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

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Step 3.1.15.2.1

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

$3 n (m \sqrt{n}) + n^{5} \sqrt{15 m} - n^{3} \sqrt{15 m n} - 1 \cdot 5 (n^{1} n^{6})$

Step 3.1.15.2.2

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

$3 n (m \sqrt{n}) + n^{5} \sqrt{15 m} - n^{3} \sqrt{15 m n} - 1 \cdot 5 n^{1 + 6}$

Step 3.1.15.3

Add $1$ and $6$ .

$3 n (m \sqrt{n}) + n^{5} \sqrt{15 m} - n^{3} \sqrt{15 m n} - 1 \cdot 5 n^{7}$

Step 3.1.16

Multiply $- 1$ by $5$ .

$3 n (m \sqrt{n}) + n^{5} \sqrt{15 m} - n^{3} \sqrt{15 m n} - 5 n^{7}$

Step 3.2

Simplify the expression.

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Step 3.2.1

Move $n$ .

$3 m n \sqrt{n} + n^{5} \sqrt{15 m} - n^{3} \sqrt{15 m n} - 5 n^{7}$

Step 3.2.2

Move $- n^{3} \sqrt{15 m n}$ .

$3 m n \sqrt{n} + n^{5} \sqrt{15 m} - 5 n^{7} - n^{3} \sqrt{15 m n}$

Step 3.2.3

Move $3 m n \sqrt{n}$ .

$n^{5} \sqrt{15 m} - 5 n^{7} + 3 m n \sqrt{n} - n^{3} \sqrt{15 m n}$

Step 3.2.4

Reorder $n^{5} \sqrt{15 m}$ and $- 5 n^{7}$ .

$- 5 n^{7} + n^{5} \sqrt{15 m} + 3 m n \sqrt{n} - n^{3} \sqrt{15 m n}$