Finite Math Examples

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

Step 1

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

Tap for more steps...

Step 1.1

Apply the distributive property.

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

Step 1.2

Apply the distributive property.

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

Step 1.3

Apply the distributive property.

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

Step 2

Simplify and combine like terms.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Multiply $5$ by $5$ .

$25 + 5 (- i \sqrt{3}) + i \sqrt{3} \cdot 5 + i \sqrt{3} (- i \sqrt{3})$

Step 2.1.2

Multiply $- 1$ by $5$ .

$25 - 5 (i \sqrt{3}) + i \sqrt{3} \cdot 5 + i \sqrt{3} (- i \sqrt{3})$

Step 2.1.3

Move $5$ to the left of $i \sqrt{3}$ .

$25 - 5 i \sqrt{3} + 5 \cdot (i \sqrt{3}) + i \sqrt{3} (- i \sqrt{3})$

Step 2.1.4

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

Tap for more steps...

Step 2.1.4.1

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

$25 - 5 i \sqrt{3} + 5 i \sqrt{3} + \sqrt{3} (- (i^{1} i) \sqrt{3})$

Step 2.1.4.2

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

$25 - 5 i \sqrt{3} + 5 i \sqrt{3} + \sqrt{3} (- (i^{1} i^{1}) \sqrt{3})$

Step 2.1.4.3

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

$25 - 5 i \sqrt{3} + 5 i \sqrt{3} + \sqrt{3} (- i^{1 + 1} \sqrt{3})$

Step 2.1.4.4

Add $1$ and $1$ .

$25 - 5 i \sqrt{3} + 5 i \sqrt{3} + \sqrt{3} (- i^{2} \sqrt{3})$

Step 2.1.4.5

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

$25 - 5 i \sqrt{3} + 5 i \sqrt{3} - i^{2} ({\sqrt{3}}^{1} \sqrt{3})$

Step 2.1.4.6

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

$25 - 5 i \sqrt{3} + 5 i \sqrt{3} - i^{2} ({\sqrt{3}}^{1} {\sqrt{3}}^{1})$

Step 2.1.4.7

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

$25 - 5 i \sqrt{3} + 5 i \sqrt{3} - i^{2} {\sqrt{3}}^{1 + 1}$

Step 2.1.4.8

Add $1$ and $1$ .

$25 - 5 i \sqrt{3} + 5 i \sqrt{3} - i^{2} {\sqrt{3}}^{2}$

Step 2.1.5

Rewrite $i^{2}$ as $- 1$ .

$25 - 5 i \sqrt{3} + 5 i \sqrt{3} - - 1 {\sqrt{3}}^{2}$

Step 2.1.6

Multiply $- 1$ by $- 1$ .

$25 - 5 i \sqrt{3} + 5 i \sqrt{3} + 1 {\sqrt{3}}^{2}$

Step 2.1.7

Multiply ${\sqrt{3}}^{2}$ by $1$ .

$25 - 5 i \sqrt{3} + 5 i \sqrt{3} + {\sqrt{3}}^{2}$

Step 2.1.8

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

Tap for more steps...

Step 2.1.8.1

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

$25 - 5 i \sqrt{3} + 5 i \sqrt{3} + {(3^{\frac{1}{2}})}^{2}$

Step 2.1.8.2

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

$25 - 5 i \sqrt{3} + 5 i \sqrt{3} + 3^{\frac{1}{2} \cdot 2}$

Step 2.1.8.3

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

$25 - 5 i \sqrt{3} + 5 i \sqrt{3} + 3^{\frac{2}{2}}$

Step 2.1.8.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.8.4.1

Cancel the common factor.

$25 - 5 i \sqrt{3} + 5 i \sqrt{3} + 3^{\frac{2}{2}}$

Step 2.1.8.4.2

Rewrite the expression.

$25 - 5 i \sqrt{3} + 5 i \sqrt{3} + 3^{1}$

Step 2.1.8.5

Evaluate the exponent.

$25 - 5 i \sqrt{3} + 5 i \sqrt{3} + 3$

Step 2.2

Add $25$ and $3$ .

$- 5 i \sqrt{3} + 5 i \sqrt{3} + 28$

Step 2.3

Add $- 5 i \sqrt{3}$ and $5 i \sqrt{3}$ .

$0 + 28$

Step 2.4

Add $0$ and $28$ .

$28$