Precalculus Examples

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

Step 1

Expand $(2 \sqrt{3 m} + 3 \sqrt{5 n}) (2 \sqrt{3 m} - 3 \sqrt{5 n})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

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Step 2.1

Simplify each term.

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Step 2.1.1

Multiply $2 \sqrt{3 m} (2 \sqrt{3 m})$ .

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Step 2.1.1.1

Multiply $2$ by $2$ .

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

Step 2.1.1.2

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

$4 ({\sqrt{3 m}}^{1} \sqrt{3 m}) + 2 \sqrt{3 m} (- 3 \sqrt{5 n}) + 3 \sqrt{5 n} (2 \sqrt{3 m}) + 3 \sqrt{5 n} (- 3 \sqrt{5 n})$

Step 2.1.1.3

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

$4 ({\sqrt{3 m}}^{1} {\sqrt{3 m}}^{1}) + 2 \sqrt{3 m} (- 3 \sqrt{5 n}) + 3 \sqrt{5 n} (2 \sqrt{3 m}) + 3 \sqrt{5 n} (- 3 \sqrt{5 n})$

Step 2.1.1.4

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

$4 {\sqrt{3 m}}^{1 + 1} + 2 \sqrt{3 m} (- 3 \sqrt{5 n}) + 3 \sqrt{5 n} (2 \sqrt{3 m}) + 3 \sqrt{5 n} (- 3 \sqrt{5 n})$

Step 2.1.1.5

Add $1$ and $1$ .

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

Step 2.1.2

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

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Step 2.1.2.1

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

$4 {({(3 m)}^{\frac{1}{2}})}^{2} + 2 \sqrt{3 m} (- 3 \sqrt{5 n}) + 3 \sqrt{5 n} (2 \sqrt{3 m}) + 3 \sqrt{5 n} (- 3 \sqrt{5 n})$

Step 2.1.2.2

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

$4 {(3 m)}^{\frac{1}{2} \cdot 2} + 2 \sqrt{3 m} (- 3 \sqrt{5 n}) + 3 \sqrt{5 n} (2 \sqrt{3 m}) + 3 \sqrt{5 n} (- 3 \sqrt{5 n})$

Step 2.1.2.3

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

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

Step 2.1.2.4

Cancel the common factor of $2$ .

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Step 2.1.2.4.1

Cancel the common factor.

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

Step 2.1.2.4.2

Rewrite the expression.

$4 {(3 m)}^{1} + 2 \sqrt{3 m} (- 3 \sqrt{5 n}) + 3 \sqrt{5 n} (2 \sqrt{3 m}) + 3 \sqrt{5 n} (- 3 \sqrt{5 n})$

Step 2.1.2.5

Simplify.

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

Step 2.1.3

Multiply $3$ by $4$ .

$12 m + 2 \sqrt{3 m} (- 3 \sqrt{5 n}) + 3 \sqrt{5 n} (2 \sqrt{3 m}) + 3 \sqrt{5 n} (- 3 \sqrt{5 n})$

Step 2.1.4

Multiply $2 \sqrt{3 m} (- 3 \sqrt{5 n})$ .

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Step 2.1.4.1

Multiply $- 3$ by $2$ .

$12 m - 6 \sqrt{3 m} \sqrt{5 n} + 3 \sqrt{5 n} (2 \sqrt{3 m}) + 3 \sqrt{5 n} (- 3 \sqrt{5 n})$

Step 2.1.4.2

Combine using the product rule for radicals.

$12 m - 6 \sqrt{5 n (3 m)} + 3 \sqrt{5 n} (2 \sqrt{3 m}) + 3 \sqrt{5 n} (- 3 \sqrt{5 n})$

Step 2.1.4.3

Multiply $3$ by $5$ .

$12 m - 6 \sqrt{15 n m} + 3 \sqrt{5 n} (2 \sqrt{3 m}) + 3 \sqrt{5 n} (- 3 \sqrt{5 n})$

Step 2.1.5

Multiply $3 \sqrt{5 n} (2 \sqrt{3 m})$ .

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Step 2.1.5.1

Multiply $2$ by $3$ .

$12 m - 6 \sqrt{15 n m} + 6 \sqrt{5 n} \sqrt{3 m} + 3 \sqrt{5 n} (- 3 \sqrt{5 n})$

Step 2.1.5.2

Combine using the product rule for radicals.

$12 m - 6 \sqrt{15 n m} + 6 \sqrt{3 m (5 n)} + 3 \sqrt{5 n} (- 3 \sqrt{5 n})$

Step 2.1.5.3

Multiply $5$ by $3$ .

$12 m - 6 \sqrt{15 n m} + 6 \sqrt{15 m n} + 3 \sqrt{5 n} (- 3 \sqrt{5 n})$

Step 2.1.6

Multiply $3 \sqrt{5 n} (- 3 \sqrt{5 n})$ .

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Step 2.1.6.1

Multiply $- 3$ by $3$ .

$12 m - 6 \sqrt{15 n m} + 6 \sqrt{15 m n} - 9 \sqrt{5 n} \sqrt{5 n}$

Step 2.1.6.2

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

$12 m - 6 \sqrt{15 n m} + 6 \sqrt{15 m n} - 9 ({\sqrt{5 n}}^{1} \sqrt{5 n})$

Step 2.1.6.3

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

$12 m - 6 \sqrt{15 n m} + 6 \sqrt{15 m n} - 9 ({\sqrt{5 n}}^{1} {\sqrt{5 n}}^{1})$

Step 2.1.6.4

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

$12 m - 6 \sqrt{15 n m} + 6 \sqrt{15 m n} - 9 {\sqrt{5 n}}^{1 + 1}$

Step 2.1.6.5

Add $1$ and $1$ .

$12 m - 6 \sqrt{15 n m} + 6 \sqrt{15 m n} - 9 {\sqrt{5 n}}^{2}$

Step 2.1.7

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

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Step 2.1.7.1

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

$12 m - 6 \sqrt{15 n m} + 6 \sqrt{15 m n} - 9 {({(5 n)}^{\frac{1}{2}})}^{2}$

Step 2.1.7.2

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

$12 m - 6 \sqrt{15 n m} + 6 \sqrt{15 m n} - 9 {(5 n)}^{\frac{1}{2} \cdot 2}$

Step 2.1.7.3

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

$12 m - 6 \sqrt{15 n m} + 6 \sqrt{15 m n} - 9 {(5 n)}^{\frac{2}{2}}$

Step 2.1.7.4

Cancel the common factor of $2$ .

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Step 2.1.7.4.1

Cancel the common factor.

$12 m - 6 \sqrt{15 n m} + 6 \sqrt{15 m n} - 9 {(5 n)}^{\frac{2}{2}}$

Step 2.1.7.4.2

Rewrite the expression.

$12 m - 6 \sqrt{15 n m} + 6 \sqrt{15 m n} - 9 {(5 n)}^{1}$

Step 2.1.7.5

Simplify.

$12 m - 6 \sqrt{15 n m} + 6 \sqrt{15 m n} - 9 (5 n)$

Step 2.1.8

Multiply $5$ by $- 9$ .

$12 m - 6 \sqrt{15 n m} + 6 \sqrt{15 m n} - 45 n$

Step 2.2

Add $- 6 \sqrt{15 n m}$ and $6 \sqrt{15 m n}$ .

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Step 2.2.1

Move $n$ .

$12 m - 6 \sqrt{15 m n} + 6 \sqrt{15 m n} - 45 n$

Step 2.2.2

Add $- 6 \sqrt{15 m n}$ and $6 \sqrt{15 m n}$ .

$12 m + 0 - 45 n$

Step 2.3

Add $12 m$ and $0$ .

$12 m - 45 n$