Simplify ( square root of 7+i square root of 7)/((1-i square root of 3)(2-2i))

Solution:

Step:1

Multiply the complex fraction $\frac{\sqrt{7}+i\sqrt{7}}{(1-i\sqrt{3})(2-2i)}$ by the conjugate of the denominator $\frac{1+i\sqrt{3}}{1+i\sqrt{3}}$ to rationalize the denominator:

$$\frac{\sqrt{7}+i\sqrt{7}}{(1-i\sqrt{3})(2-2i)}\cdot \frac{1+i\sqrt{3}}{1+i\sqrt{3}}$$

Step:2

Combine the fractions into a single expression:

$$\frac{(\sqrt{7}+i\sqrt{7})(1+i\sqrt{3})}{(1-i\sqrt{3})(2-2i)(1+i\sqrt{3})}$$

Step:2.1

Rearrange the denominator:

$$\frac{(\sqrt{7}+i\sqrt{7})(1+i\sqrt{3})}{(2-2i)((1-i\sqrt{3})(1+i\sqrt{3}))}$$

Step:2.2

Expand the denominator using the difference of squares formula:

$$\frac{(\sqrt{7}+i\sqrt{7})(1+i\sqrt{3})}{(2-2i)(1-(i\sqrt{3}{)}^2)}$$

Step:2.3

Simplify the denominator:

$$\frac{(\sqrt{7}+i\sqrt{7})(1+i\sqrt{3})}{(2-2i)\cdot 4}$$

Step:3

Expand the numerator using the distributive property (FOIL method):

$$\frac{\sqrt{7}(1+i\sqrt{3})+i\sqrt{7}(1+i\sqrt{3})}{(2-2i)\cdot 4}$$

Step:4

Simplify each term in the numerator:

Step:4.1

Multiply $\sqrt{7}$ by $1$ and $i\sqrt{3}$:

$$\frac{\sqrt{7}+\sqrt{7}i\sqrt{3}+i\sqrt{7}+i\sqrt{7}i\sqrt{3}}{(2-2i)\cdot 4}$$

Step:4.2

Combine like terms and simplify the expression:

$$\frac{\sqrt{7}+\sqrt{21}i+i\sqrt{7}-\sqrt{21}}{(2-2i)\cdot 4}$$

Step:4.3

Factor out the common factor of $-1$ from the numerator:

$$-\frac{\sqrt{21}-\sqrt{7}+\sqrt{21}i+i\sqrt{7}}{4(2-2i)}$$

Step:4.4

Simplify the expression:

$$-\frac{\sqrt{21}-\sqrt{7}+(\sqrt{21}+\sqrt{7})i}{8-8i}$$

Knowledge Notes:

1. Complex Numbers: A complex number is a number of the form $a+bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit with the property $i^2=-1$.

2. Rationalizing the Denominator: This process involves eliminating the imaginary part from the denominator of a complex fraction by multiplying the numerator and denominator by the conjugate of the denominator.

3. Conjugate of a Complex Number: The conjugate of a complex number $a+bi$ is $a-bi$. Multiplying a complex number by its conjugate results in a real number.

4. Distributive Property (FOIL Method): This property is used to expand the product of two binomials. FOIL stands for First, Outer, Inner, Last, which are the terms that are multiplied together.

5. Difference of Squares: This is a pattern that allows us to factor expressions of the form $a^2-b^2$ into $(a+b)(a-b)$.

6. Simplifying Complex Expressions: This involves combining like terms, factoring, and reducing expressions to their simplest form.