Solve in Terms of the Arbitrary Variable u u=(1+i square root of 3) v=(1+2i square root of 3)

$u = (1 + i \sqrt{3})$ $v = (1 + 2 i \sqrt{3})$

Step 1

Remove parentheses.

$v = 1 + 2 (\frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}) \sqrt{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Step 2

Simplify $1 + 2 (\frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}) \sqrt{3}$ .

Simplify each term.

Apply the distributive property.

$v = 1 + (2 (\frac{u \sqrt{3}}{3}) + 2 (- \frac{\sqrt{3}}{3})) \sqrt{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Combine $2$ and $\frac{u \sqrt{3}}{3}$ .

$v = 1 + (\frac{2 (u \sqrt{3})}{3} + 2 (- \frac{\sqrt{3}}{3})) \sqrt{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Multiply $2 (- \frac{\sqrt{3}}{3})$ .

Multiply $- 1$ by $2$ .

$v = 1 + (\frac{2 (u \sqrt{3})}{3} - 2 \frac{\sqrt{3}}{3}) \sqrt{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Combine $- 2$ and $\frac{\sqrt{3}}{3}$ .

$v = 1 + (\frac{2 (u \sqrt{3})}{3} + \frac{- 2 \sqrt{3}}{3}) \sqrt{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

$v = 1 + (\frac{2 (u \sqrt{3})}{3} + \frac{- 2 \sqrt{3}}{3}) \sqrt{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Move the negative in front of the fraction.

$v = 1 + (\frac{2 u \sqrt{3}}{3} - \frac{2 \sqrt{3}}{3}) \sqrt{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Apply the distributive property.

$v = 1 + \frac{2 u \sqrt{3}}{3} \cdot \sqrt{3} - \frac{2 \sqrt{3}}{3} \cdot \sqrt{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Multiply $\frac{2 u \sqrt{3}}{3} \sqrt{3}$ .

Combine $\frac{2 u \sqrt{3}}{3}$ and $\sqrt{3}$ .

$v = 1 + \frac{2 u \sqrt{3} \sqrt{3}}{3} - \frac{2 \sqrt{3}}{3} \cdot \sqrt{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

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

$v = 1 + \frac{2 u (\sqrt{3} \sqrt{3})}{3} - \frac{2 \sqrt{3}}{3} \cdot \sqrt{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

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

$v = 1 + \frac{2 u (\sqrt{3} \sqrt{3})}{3} - \frac{2 \sqrt{3}}{3} \cdot \sqrt{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

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

$v = 1 + \frac{2 u {\sqrt{3}}^{1 + 1}}{3} - \frac{2 \sqrt{3}}{3} \cdot \sqrt{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Add $1$ and $1$ .

$v = 1 + \frac{2 u {\sqrt{3}}^{2}}{3} - \frac{2 \sqrt{3}}{3} \cdot \sqrt{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

$v = 1 + \frac{2 u {\sqrt{3}}^{2}}{3} - \frac{2 \sqrt{3}}{3} \cdot \sqrt{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Multiply $- \frac{2 \sqrt{3}}{3} \sqrt{3}$ .

Combine $\sqrt{3}$ and $\frac{2 \sqrt{3}}{3}$ .

$v = 1 + \frac{2 u {\sqrt{3}}^{2}}{3} - \frac{\sqrt{3} (2 \sqrt{3})}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

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

$v = 1 + \frac{2 u {\sqrt{3}}^{2}}{3} - \frac{2 (\sqrt{3} \sqrt{3})}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

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

$v = 1 + \frac{2 u {\sqrt{3}}^{2}}{3} - \frac{2 (\sqrt{3} \sqrt{3})}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

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

$v = 1 + \frac{2 u {\sqrt{3}}^{2}}{3} - \frac{2 {\sqrt{3}}^{1 + 1}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Add $1$ and $1$ .

$v = 1 + \frac{2 u {\sqrt{3}}^{2}}{3} - \frac{2 {\sqrt{3}}^{2}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

$v = 1 + \frac{2 u {\sqrt{3}}^{2}}{3} - \frac{2 {\sqrt{3}}^{2}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Simplify each term.

Simplify the numerator.

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

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

$v = 1 + \frac{2 u {(3^{\frac{1}{2}})}^{2}}{3} - \frac{2 {\sqrt{3}}^{2}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

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

$v = 1 + \frac{2 u \cdot 3^{\frac{1}{2} \cdot 2}}{3} - \frac{2 {\sqrt{3}}^{2}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

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

$v = 1 + \frac{2 u \cdot 3^{\frac{2}{2}}}{3} - \frac{2 {\sqrt{3}}^{2}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Cancel the common factor of $2$ .

Cancel the common factor.

$v = 1 + \frac{2 u \cdot 3^{\frac{2}{2}}}{3} - \frac{2 {\sqrt{3}}^{2}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Rewrite the expression.

$v = 1 + \frac{2 u \cdot 3}{3} - \frac{2 {\sqrt{3}}^{2}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

$v = 1 + \frac{2 u \cdot 3}{3} - \frac{2 {\sqrt{3}}^{2}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Evaluate the exponent.

$v = 1 + \frac{2 u \cdot 3}{3} - \frac{2 {\sqrt{3}}^{2}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

$v = 1 + \frac{2 u \cdot 3}{3} - \frac{2 {\sqrt{3}}^{2}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Multiply $3$ by $2$ .

$v = 1 + \frac{6 u}{3} - \frac{2 {\sqrt{3}}^{2}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

$v = 1 + \frac{6 u}{3} - \frac{2 {\sqrt{3}}^{2}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Cancel the common factor of $6$ and $3$ .

Factor $3$ out of $6 u$ .

$v = 1 + \frac{3 (2 u)}{3} - \frac{2 {\sqrt{3}}^{2}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

$v = 1 + \frac{3 (2 u)}{3 (1)} - \frac{2 {\sqrt{3}}^{2}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Cancel the common factor.

$v = 1 + \frac{3 (2 u)}{3 \cdot 1} - \frac{2 {\sqrt{3}}^{2}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Rewrite the expression.

$v = 1 + \frac{2 u}{1} - \frac{2 {\sqrt{3}}^{2}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Divide $2 u$ by $1$ .

$v = 1 + 2 u - \frac{2 {\sqrt{3}}^{2}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

$v = 1 + 2 u - \frac{2 {\sqrt{3}}^{2}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

$v = 1 + 2 u - \frac{2 {\sqrt{3}}^{2}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

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

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

$v = 1 + 2 u - \frac{2 {(3^{\frac{1}{2}})}^{2}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

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

$v = 1 + 2 u - \frac{2 \cdot 3^{\frac{1}{2} \cdot 2}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

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

$v = 1 + 2 u - \frac{2 \cdot 3^{\frac{2}{2}}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Cancel the common factor of $2$ .

Cancel the common factor.

$v = 1 + 2 u - \frac{2 \cdot 3^{\frac{2}{2}}}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Rewrite the expression.

$v = 1 + 2 u - \frac{2 \cdot 3}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

$v = 1 + 2 u - \frac{2 \cdot 3}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Evaluate the exponent.

$v = 1 + 2 u - \frac{2 \cdot 3}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

$v = 1 + 2 u - \frac{2 \cdot 3}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Multiply $2$ by $3$ .

$v = 1 + 2 u - \frac{6}{3}$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Divide $6$ by $3$ .

$v = 1 + 2 u - 1 \cdot 2$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Multiply $- 1$ by $2$ .

$v = 1 + 2 u - 2$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

$v = 1 + 2 u - 2$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

$v = 1 + 2 u - 2$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Subtract $2$ from $1$ .

$v = 2 u - 1$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

$v = 2 u - 1$

$i = \frac{u \sqrt{3}}{3} - \frac{\sqrt{3}}{3}$

Solve in Terms of the Arbitrary Variable u u=(1+i square root of 3) v=(1+2i square root of 3)

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