Finite Math Examples

$(2 \sqrt{3} - 5 \sqrt{10}) (3 \sqrt{3} + \sqrt{10})$

Step 1

Expand $(2 \sqrt{3} - 5 \sqrt{10}) (3 \sqrt{3} + \sqrt{10})$ using the FOIL Method.

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Step 1.1

Apply the distributive property.

$2 \sqrt{3} (3 \sqrt{3} + \sqrt{10}) - 5 \sqrt{10} (3 \sqrt{3} + \sqrt{10})$

Step 1.2

Apply the distributive property.

$2 \sqrt{3} (3 \sqrt{3}) + 2 \sqrt{3} \sqrt{10} - 5 \sqrt{10} (3 \sqrt{3} + \sqrt{10})$

Step 1.3

Apply the distributive property.

$2 \sqrt{3} (3 \sqrt{3}) + 2 \sqrt{3} \sqrt{10} - 5 \sqrt{10} (3 \sqrt{3}) - 5 \sqrt{10} \sqrt{10}$

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

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Step 2.1.1.1

Multiply $3$ by $2$ .

$6 \sqrt{3} \sqrt{3} + 2 \sqrt{3} \sqrt{10} - 5 \sqrt{10} (3 \sqrt{3}) - 5 \sqrt{10} \sqrt{10}$

Step 2.1.1.2

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

$6 ({\sqrt{3}}^{1} \sqrt{3}) + 2 \sqrt{3} \sqrt{10} - 5 \sqrt{10} (3 \sqrt{3}) - 5 \sqrt{10} \sqrt{10}$

Step 2.1.1.3

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

$6 ({\sqrt{3}}^{1} {\sqrt{3}}^{1}) + 2 \sqrt{3} \sqrt{10} - 5 \sqrt{10} (3 \sqrt{3}) - 5 \sqrt{10} \sqrt{10}$

Step 2.1.1.4

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

$6 {\sqrt{3}}^{1 + 1} + 2 \sqrt{3} \sqrt{10} - 5 \sqrt{10} (3 \sqrt{3}) - 5 \sqrt{10} \sqrt{10}$

Step 2.1.1.5

Add $1$ and $1$ .

$6 {\sqrt{3}}^{2} + 2 \sqrt{3} \sqrt{10} - 5 \sqrt{10} (3 \sqrt{3}) - 5 \sqrt{10} \sqrt{10}$

Step 2.1.2

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

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Step 2.1.2.1

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

$6 {(3^{\frac{1}{2}})}^{2} + 2 \sqrt{3} \sqrt{10} - 5 \sqrt{10} (3 \sqrt{3}) - 5 \sqrt{10} \sqrt{10}$

Step 2.1.2.2

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

$6 \cdot 3^{\frac{1}{2} \cdot 2} + 2 \sqrt{3} \sqrt{10} - 5 \sqrt{10} (3 \sqrt{3}) - 5 \sqrt{10} \sqrt{10}$

Step 2.1.2.3

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

$6 \cdot 3^{\frac{2}{2}} + 2 \sqrt{3} \sqrt{10} - 5 \sqrt{10} (3 \sqrt{3}) - 5 \sqrt{10} \sqrt{10}$

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.

$6 \cdot 3^{\frac{2}{2}} + 2 \sqrt{3} \sqrt{10} - 5 \sqrt{10} (3 \sqrt{3}) - 5 \sqrt{10} \sqrt{10}$

Step 2.1.2.4.2

Rewrite the expression.

$6 \cdot 3^{1} + 2 \sqrt{3} \sqrt{10} - 5 \sqrt{10} (3 \sqrt{3}) - 5 \sqrt{10} \sqrt{10}$

Step 2.1.2.5

Evaluate the exponent.

$6 \cdot 3 + 2 \sqrt{3} \sqrt{10} - 5 \sqrt{10} (3 \sqrt{3}) - 5 \sqrt{10} \sqrt{10}$

Step 2.1.3

Multiply $6$ by $3$ .

$18 + 2 \sqrt{3} \sqrt{10} - 5 \sqrt{10} (3 \sqrt{3}) - 5 \sqrt{10} \sqrt{10}$

Step 2.1.4

Multiply $2 \sqrt{3} \sqrt{10}$ .

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Step 2.1.4.1

Combine using the product rule for radicals.

$18 + 2 \sqrt{10 \cdot 3} - 5 \sqrt{10} (3 \sqrt{3}) - 5 \sqrt{10} \sqrt{10}$

Step 2.1.4.2

Multiply $10$ by $3$ .

$18 + 2 \sqrt{30} - 5 \sqrt{10} (3 \sqrt{3}) - 5 \sqrt{10} \sqrt{10}$

Step 2.1.5

Multiply $- 5 \sqrt{10} (3 \sqrt{3})$ .

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Step 2.1.5.1

Multiply $3$ by $- 5$ .

$18 + 2 \sqrt{30} - 15 \sqrt{10} \sqrt{3} - 5 \sqrt{10} \sqrt{10}$

Step 2.1.5.2

Combine using the product rule for radicals.

$18 + 2 \sqrt{30} - 15 \sqrt{3 \cdot 10} - 5 \sqrt{10} \sqrt{10}$

Step 2.1.5.3

Multiply $3$ by $10$ .

$18 + 2 \sqrt{30} - 15 \sqrt{30} - 5 \sqrt{10} \sqrt{10}$

Step 2.1.6

Multiply $- 5 \sqrt{10} \sqrt{10}$ .

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Step 2.1.6.1

Raise $\sqrt{10}$ to the power of $1$ .

$18 + 2 \sqrt{30} - 15 \sqrt{30} - 5 ({\sqrt{10}}^{1} \sqrt{10})$

Step 2.1.6.2

Raise $\sqrt{10}$ to the power of $1$ .

$18 + 2 \sqrt{30} - 15 \sqrt{30} - 5 ({\sqrt{10}}^{1} {\sqrt{10}}^{1})$

Step 2.1.6.3

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

$18 + 2 \sqrt{30} - 15 \sqrt{30} - 5 {\sqrt{10}}^{1 + 1}$

Step 2.1.6.4

Add $1$ and $1$ .

$18 + 2 \sqrt{30} - 15 \sqrt{30} - 5 {\sqrt{10}}^{2}$

Step 2.1.7

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 2.1.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$18 + 2 \sqrt{30} - 15 \sqrt{30} - 5 {(10^{\frac{1}{2}})}^{2}$

Step 2.1.7.2

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

$18 + 2 \sqrt{30} - 15 \sqrt{30} - 5 \cdot 10^{\frac{1}{2} \cdot 2}$

Step 2.1.7.3

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

$18 + 2 \sqrt{30} - 15 \sqrt{30} - 5 \cdot 10^{\frac{2}{2}}$

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.

$18 + 2 \sqrt{30} - 15 \sqrt{30} - 5 \cdot 10^{\frac{2}{2}}$

Step 2.1.7.4.2

Rewrite the expression.

$18 + 2 \sqrt{30} - 15 \sqrt{30} - 5 \cdot 10^{1}$

Step 2.1.7.5

Evaluate the exponent.

$18 + 2 \sqrt{30} - 15 \sqrt{30} - 5 \cdot 10$

Step 2.1.8

Multiply $- 5$ by $10$ .

$18 + 2 \sqrt{30} - 15 \sqrt{30} - 50$

Step 2.2

Subtract $50$ from $18$ .

$- 32 + 2 \sqrt{30} - 15 \sqrt{30}$

Step 2.3

Subtract $15 \sqrt{30}$ from $2 \sqrt{30}$ .

$- 32 - 13 \sqrt{30}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$- 32 - 13 \sqrt{30}$

Decimal Form:

$- 103.20393247 \dots$