Basic Math Examples

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

Step 1

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

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Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

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 $2$ by $2$ .

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

Step 2.1.2

Multiply $- 5$ by $2$ .

$4 - 10 \sqrt{3} + 5 \sqrt{3} \cdot 2 + 5 \sqrt{3} (- 5 \sqrt{3})$

Step 2.1.3

Multiply $2$ by $5$ .

$4 - 10 \sqrt{3} + 10 \sqrt{3} + 5 \sqrt{3} (- 5 \sqrt{3})$

Step 2.1.4

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

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Step 2.1.4.1

Multiply $- 5$ by $5$ .

$4 - 10 \sqrt{3} + 10 \sqrt{3} - 25 \sqrt{3} \sqrt{3}$

Step 2.1.4.2

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

$4 - 10 \sqrt{3} + 10 \sqrt{3} - 25 ({\sqrt{3}}^{1} \sqrt{3})$

Step 2.1.4.3

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

$4 - 10 \sqrt{3} + 10 \sqrt{3} - 25 ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 2.1.4.4

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

$4 - 10 \sqrt{3} + 10 \sqrt{3} - 25 {\sqrt{3}}^{1 + 1}$

Step 2.1.4.5

Add $1$ and $1$ .

$4 - 10 \sqrt{3} + 10 \sqrt{3} - 25 {\sqrt{3}}^{2}$

Step 2.1.5

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

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Step 2.1.5.1

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

$4 - 10 \sqrt{3} + 10 \sqrt{3} - 25 {(3^{\frac{1}{2}})}^{2}$

Step 2.1.5.2

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

$4 - 10 \sqrt{3} + 10 \sqrt{3} - 25 \cdot 3^{\frac{1}{2} \cdot 2}$

Step 2.1.5.3

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

$4 - 10 \sqrt{3} + 10 \sqrt{3} - 25 \cdot 3^{\frac{2}{2}}$

Step 2.1.5.4

Cancel the common factor of $2$ .

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Step 2.1.5.4.1

Cancel the common factor.

$4 - 10 \sqrt{3} + 10 \sqrt{3} - 25 \cdot 3^{\frac{2}{2}}$

Step 2.1.5.4.2

Rewrite the expression.

$4 - 10 \sqrt{3} + 10 \sqrt{3} - 25 \cdot 3^{1}$

Step 2.1.5.5

Evaluate the exponent.

$4 - 10 \sqrt{3} + 10 \sqrt{3} - 25 \cdot 3$

Step 2.1.6

Multiply $- 25$ by $3$ .

$4 - 10 \sqrt{3} + 10 \sqrt{3} - 75$

Step 2.2

Subtract $75$ from $4$ .

$- 71 - 10 \sqrt{3} + 10 \sqrt{3}$

Step 2.3

Add $- 10 \sqrt{3}$ and $10 \sqrt{3}$ .

$- 71 + 0$

Step 2.4

Add $- 71$ and $0$ .

$- 71$