Simplify (( square root of 5)/5-( square root of 2)/5i)^2

${(\frac{\sqrt{5}}{5} - \frac{\sqrt{2}}{5} i)}^{2}$

Step 1

Simplify terms.

Combine $i$ and $\frac{\sqrt{2}}{5}$ .

${(\frac{\sqrt{5}}{5} - \frac{i \sqrt{2}}{5})}^{2}$

Rewrite ${(\frac{\sqrt{5}}{5} - \frac{i \sqrt{2}}{5})}^{2}$ as $(\frac{\sqrt{5}}{5} - \frac{i \sqrt{2}}{5}) (\frac{\sqrt{5}}{5} - \frac{i \sqrt{2}}{5})$ .

$(\frac{\sqrt{5}}{5} - \frac{i \sqrt{2}}{5}) (\frac{\sqrt{5}}{5} - \frac{i \sqrt{2}}{5})$

Step 2

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

Apply the distributive property.

$\frac{\sqrt{5}}{5} (\frac{\sqrt{5}}{5} - \frac{i \sqrt{2}}{5}) - \frac{i \sqrt{2}}{5} (\frac{\sqrt{5}}{5} - \frac{i \sqrt{2}}{5})$

Apply the distributive property.

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

Apply the distributive property.

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

Step 3

Simplify and combine like terms.

Simplify each term.

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

$\frac{{\sqrt{5}}^{2}}{5 \cdot 5} + \frac{\sqrt{5}}{5} (- \frac{i \sqrt{2}}{5}) - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{5}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Multiply $5$ by $5$ .

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

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

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

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

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

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

$\frac{5}{25} + \frac{\sqrt{5}}{5} (- \frac{i \sqrt{2}}{5}) - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{5}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Cancel the common factor of $5$ and $25$ .

Factor $5$ out of $5$ .

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

Cancel the common factors.

Factor $5$ out of $25$ .

$\frac{5 \cdot 1}{5 \cdot 5} + \frac{\sqrt{5}}{5} (- \frac{i \sqrt{2}}{5}) - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{5}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Cancel the common factor.

$\frac{5 \cdot 1}{5 \cdot 5} + \frac{\sqrt{5}}{5} (- \frac{i \sqrt{2}}{5}) - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{5}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Rewrite the expression.

$\frac{1}{5} + \frac{\sqrt{5}}{5} (- \frac{i \sqrt{2}}{5}) - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{5}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Multiply $\frac{\sqrt{5}}{5} (- \frac{i \sqrt{2}}{5})$ .

Multiply $\frac{\sqrt{5}}{5}$ by $\frac{i \sqrt{2}}{5}$ .

$\frac{1}{5} - \frac{\sqrt{5} (i \sqrt{2})}{5 \cdot 5} - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{5}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Combine using the product rule for radicals.

$\frac{1}{5} - \frac{\sqrt{2 \cdot 5} i}{5 \cdot 5} - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{5}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Multiply $2$ by $5$ .

$\frac{1}{5} - \frac{\sqrt{10} i}{5 \cdot 5} - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{5}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Multiply $5$ by $5$ .

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{5}}{5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Multiply $- \frac{i \sqrt{2}}{5} \cdot \frac{\sqrt{5}}{5}$ .

Multiply $\frac{\sqrt{5}}{5}$ by $\frac{i \sqrt{2}}{5}$ .

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{5} (i \sqrt{2})}{5 \cdot 5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Combine using the product rule for radicals.

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{2 \cdot 5} i}{5 \cdot 5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Multiply $2$ by $5$ .

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{5 \cdot 5} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Multiply $5$ by $5$ .

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25} - \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$

Multiply $- \frac{i \sqrt{2}}{5} (- \frac{i \sqrt{2}}{5})$ .

Multiply $- 1$ by $- 1$ .

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25} + 1 \frac{i \sqrt{2}}{5} \frac{i \sqrt{2}}{5}$

Multiply $\frac{i \sqrt{2}}{5}$ by $1$ .

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25} + \frac{i \sqrt{2}}{5} \cdot \frac{i \sqrt{2}}{5}$

Multiply $\frac{i \sqrt{2}}{5}$ by $\frac{i \sqrt{2}}{5}$ .

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25} + \frac{i \sqrt{2} (i \sqrt{2})}{5 \cdot 5}$

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

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25} + \frac{i^{1} i \sqrt{2} \sqrt{2}}{5 \cdot 5}$

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

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25} + \frac{i^{1} i^{1} \sqrt{2} \sqrt{2}}{5 \cdot 5}$

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

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25} + \frac{i^{1 + 1} \sqrt{2} \sqrt{2}}{5 \cdot 5}$

Add $1$ and $1$ .

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25} + \frac{i^{2} \sqrt{2} \sqrt{2}}{5 \cdot 5}$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25} + \frac{i^{2} ({\sqrt{2}}^{1} \sqrt{2})}{5 \cdot 5}$

Raise $\sqrt{2}$ to the power of $1$ .

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25} + \frac{i^{2} ({\sqrt{2}}^{1} {\sqrt{2}}^{1})}{5 \cdot 5}$

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

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25} + \frac{i^{2} {\sqrt{2}}^{1 + 1}}{5 \cdot 5}$

Add $1$ and $1$ .

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25} + \frac{i^{2} {\sqrt{2}}^{2}}{5 \cdot 5}$

Multiply $5$ by $5$ .

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25} + \frac{i^{2} {\sqrt{2}}^{2}}{25}$

Simplify the numerator.

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

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25} + \frac{- {\sqrt{2}}^{2}}{25}$

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

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

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25} + \frac{- {(2^{\frac{1}{2}})}^{2}}{25}$

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

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25} + \frac{- 2^{\frac{1}{2} \cdot 2}}{25}$

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

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25} + \frac{- 2^{\frac{2}{2}}}{25}$

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25} + \frac{- 2^{\frac{2}{2}}}{25}$

Rewrite the expression.

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25} + \frac{- 2^{1}}{25}$

Evaluate the exponent.

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25} + \frac{- 1 \cdot 2}{25}$

Multiply $- 1$ by $2$ .

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25} + \frac{- 2}{25}$

Move the negative in front of the fraction.

$\frac{1}{5} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25} - \frac{2}{25}$

To write $\frac{1}{5}$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\frac{1}{5} \cdot \frac{5}{5} - \frac{2}{25} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25}$

Write each expression with a common denominator of $25$ , by multiplying each by an appropriate factor of $1$ .

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

$\frac{5}{5 \cdot 5} - \frac{2}{25} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25}$

Multiply $5$ by $5$ .

$\frac{5}{25} - \frac{2}{25} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25}$

Combine the numerators over the common denominator.

$\frac{5 - 2}{25} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25}$

Subtract $2$ from $5$ .

$\frac{3}{25} - \frac{\sqrt{10} i}{25} - \frac{\sqrt{10} i}{25}$

Subtract $\frac{\sqrt{10} i}{25}$ from $- \frac{\sqrt{10} i}{25}$ .

$\frac{3}{25} - 2 \frac{\sqrt{10} i}{25}$

Step 4

Simplify each term.

Combine $- 2$ and $\frac{\sqrt{10} i}{25}$ .

$\frac{3}{25} + \frac{- 2 (\sqrt{10} i)}{25}$

Move the negative in front of the fraction.

$\frac{3}{25} - \frac{2 \sqrt{10} i}{25}$

Simplify (( square root of 5)/5-( square root of 2)/5i)^2

Related Questions

Support LCMGCF.com by disabling your adblocker.