Simplify (1/5+( square root of 19)/10*i)^2

${(\frac{1}{5} + \frac{\sqrt{19}}{10} \cdot i)}^{2}$

Step 1

Simplify terms.

Combine $\frac{\sqrt{19}}{10}$ and $i$ .

${(\frac{1}{5} + \frac{\sqrt{19} i}{10})}^{2}$

Rewrite ${(\frac{1}{5} + \frac{\sqrt{19} i}{10})}^{2}$ as $(\frac{1}{5} + \frac{\sqrt{19} i}{10}) (\frac{1}{5} + \frac{\sqrt{19} i}{10})$ .

$(\frac{1}{5} + \frac{\sqrt{19} i}{10}) (\frac{1}{5} + \frac{\sqrt{19} i}{10})$

Step 2

Expand $(\frac{1}{5} + \frac{\sqrt{19} i}{10}) (\frac{1}{5} + \frac{\sqrt{19} i}{10})$ using the FOIL Method.

Apply the distributive property.

$\frac{1}{5} (\frac{1}{5} + \frac{\sqrt{19} i}{10}) + \frac{\sqrt{19} i}{10} (\frac{1}{5} + \frac{\sqrt{19} i}{10})$

Apply the distributive property.

$\frac{1}{5} \cdot \frac{1}{5} + \frac{1}{5} \cdot \frac{\sqrt{19} i}{10} + \frac{\sqrt{19} i}{10} (\frac{1}{5} + \frac{\sqrt{19} i}{10})$

Apply the distributive property.

$\frac{1}{5} \cdot \frac{1}{5} + \frac{1}{5} \cdot \frac{\sqrt{19} i}{10} + \frac{\sqrt{19} i}{10} \cdot \frac{1}{5} + \frac{\sqrt{19} i}{10} \cdot \frac{\sqrt{19} i}{10}$

Step 3

Simplify and combine like terms.

Simplify each term.

Multiply $\frac{1}{5} \cdot \frac{1}{5}$ .

Multiply $\frac{1}{5}$ by $\frac{1}{5}$ .

$\frac{1}{5 \cdot 5} + \frac{1}{5} \cdot \frac{\sqrt{19} i}{10} + \frac{\sqrt{19} i}{10} \cdot \frac{1}{5} + \frac{\sqrt{19} i}{10} \cdot \frac{\sqrt{19} i}{10}$

Multiply $5$ by $5$ .

$\frac{1}{25} + \frac{1}{5} \cdot \frac{\sqrt{19} i}{10} + \frac{\sqrt{19} i}{10} \cdot \frac{1}{5} + \frac{\sqrt{19} i}{10} \cdot \frac{\sqrt{19} i}{10}$

Multiply $\frac{1}{5} \cdot \frac{\sqrt{19} i}{10}$ .

Multiply $\frac{1}{5}$ by $\frac{\sqrt{19} i}{10}$ .

$\frac{1}{25} + \frac{\sqrt{19} i}{5 \cdot 10} + \frac{\sqrt{19} i}{10} \cdot \frac{1}{5} + \frac{\sqrt{19} i}{10} \cdot \frac{\sqrt{19} i}{10}$

Multiply $5$ by $10$ .

$\frac{1}{25} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{10} \cdot \frac{1}{5} + \frac{\sqrt{19} i}{10} \cdot \frac{\sqrt{19} i}{10}$

Multiply $\frac{\sqrt{19} i}{10} \cdot \frac{1}{5}$ .

Multiply $\frac{\sqrt{19} i}{10}$ by $\frac{1}{5}$ .

$\frac{1}{25} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{10 \cdot 5} + \frac{\sqrt{19} i}{10} \cdot \frac{\sqrt{19} i}{10}$

Multiply $10$ by $5$ .

$\frac{1}{25} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{10} \cdot \frac{\sqrt{19} i}{10}$

Multiply $\frac{\sqrt{19} i}{10} \cdot \frac{\sqrt{19} i}{10}$ .

Multiply $\frac{\sqrt{19} i}{10}$ by $\frac{\sqrt{19} i}{10}$ .

$\frac{1}{25} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i (\sqrt{19} i)}{10 \cdot 10}$

Raise $\sqrt{19}$ to the power of $1$ .

$\frac{1}{25} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50} + \frac{{\sqrt{19}}^{1} \sqrt{19} i i}{10 \cdot 10}$

Raise $\sqrt{19}$ to the power of $1$ .

$\frac{1}{25} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50} + \frac{{\sqrt{19}}^{1} {\sqrt{19}}^{1} i i}{10 \cdot 10}$

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

$\frac{1}{25} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50} + \frac{{\sqrt{19}}^{1 + 1} i i}{10 \cdot 10}$

Add $1$ and $1$ .

$\frac{1}{25} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50} + \frac{{\sqrt{19}}^{2} i i}{10 \cdot 10}$

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

$\frac{1}{25} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50} + \frac{{\sqrt{19}}^{2} (i^{1} i)}{10 \cdot 10}$

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

$\frac{1}{25} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50} + \frac{{\sqrt{19}}^{2} (i^{1} i^{1})}{10 \cdot 10}$

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

$\frac{1}{25} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50} + \frac{{\sqrt{19}}^{2} i^{1 + 1}}{10 \cdot 10}$

Add $1$ and $1$ .

$\frac{1}{25} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50} + \frac{{\sqrt{19}}^{2} i^{2}}{10 \cdot 10}$

Multiply $10$ by $10$ .

$\frac{1}{25} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50} + \frac{{\sqrt{19}}^{2} i^{2}}{100}$

Simplify the numerator.

Rewrite ${\sqrt{19}}^{2}$ as $19$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{19}$ as $19^{\frac{1}{2}}$ .

$\frac{1}{25} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50} + \frac{{(19^{\frac{1}{2}})}^{2} i^{2}}{100}$

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

$\frac{1}{25} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50} + \frac{19^{\frac{1}{2} \cdot 2} i^{2}}{100}$

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

$\frac{1}{25} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50} + \frac{19^{\frac{2}{2}} i^{2}}{100}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{1}{25} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50} + \frac{19^{\frac{2}{2}} i^{2}}{100}$

Rewrite the expression.

$\frac{1}{25} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50} + \frac{19^{1} i^{2}}{100}$

Evaluate the exponent.

$\frac{1}{25} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50} + \frac{19 i^{2}}{100}$

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

$\frac{1}{25} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50} + \frac{19 \cdot - 1}{100}$

Multiply $19$ by $- 1$ .

$\frac{1}{25} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50} + \frac{- 19}{100}$

Move the negative in front of the fraction.

$\frac{1}{25} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50} - \frac{19}{100}$

To write $\frac{1}{25}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{1}{25} \cdot \frac{4}{4} - \frac{19}{100} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50}$

Write each expression with a common denominator of $100$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{1}{25}$ by $\frac{4}{4}$ .

$\frac{4}{25 \cdot 4} - \frac{19}{100} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50}$

Multiply $25$ by $4$ .

$\frac{4}{100} - \frac{19}{100} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50}$

Combine the numerators over the common denominator.

$\frac{4 - 19}{100} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50}$

Subtract $19$ from $4$ .

$\frac{- 15}{100} + \frac{\sqrt{19} i}{50} + \frac{\sqrt{19} i}{50}$

Combine the numerators over the common denominator.

$\frac{- 15}{100} + \frac{2 \sqrt{19} i}{50}$

Step 4

Simplify each term.

Cancel the common factor of $- 15$ and $100$ .

Factor $5$ out of $- 15$ .

$\frac{5 (- 3)}{100} + \frac{2 \sqrt{19} i}{50}$

Cancel the common factors.

Factor $5$ out of $100$ .

$\frac{5 \cdot - 3}{5 \cdot 20} + \frac{2 \sqrt{19} i}{50}$

Cancel the common factor.

$\frac{5 \cdot - 3}{5 \cdot 20} + \frac{2 \sqrt{19} i}{50}$

Rewrite the expression.

$\frac{- 3}{20} + \frac{2 \sqrt{19} i}{50}$

Move the negative in front of the fraction.

$- \frac{3}{20} + \frac{2 \sqrt{19} i}{50}$

Cancel the common factor of $2$ and $50$ .

Factor $2$ out of $2 \sqrt{19} i$ .

$- \frac{3}{20} + \frac{2 (\sqrt{19} i)}{50}$

Cancel the common factors.

Factor $2$ out of $50$ .

$- \frac{3}{20} + \frac{2 (\sqrt{19} i)}{2 \cdot 25}$

Cancel the common factor.

$- \frac{3}{20} + \frac{2 (\sqrt{19} i)}{2 \cdot 25}$

Rewrite the expression.

$- \frac{3}{20} + \frac{\sqrt{19} i}{25}$

Simplify (1/5+( square root of 19)/10*i)^2

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