Simplify ( square root of 7+i square root of 7)/((1-i square root of 3)(2-2i))
The given problem is asking to simplify a complex fraction involving complex numbers in both the numerator and denominator. Specifically, the numerator contains the expression $sqrt(7) + i sqrt(7)$, which involves the square root of 7 and an imaginary part that is also the square root of 7 times the imaginary unit $i$. The denominator is composed of two factors: $(1 - i sqrt(3))$and $(2 - 2i)$. Both factors in the denominator are binomials with real and imaginary parts.
To complete the simplification, one would need to perform operations such as multiplication, addition, and subtraction with complex numbers. It also might require the use of the complex conjugate to eliminate the imaginary parts from the denominator of the fraction, thereby simplifying the complex fraction to a form where it has a real denominator. The result should be a simplified expression where both the real and imaginary parts are rationalized, and the expression can be written in the standard form $a + bi$, where $a$and $b$are real numbers and $i$is the imaginary unit.
$\frac{\sqrt{7} + i \sqrt{7}}{\left(\right. 1 - i \sqrt{3} \left.\right) \left(\right. 2 - 2 i \left.\right)}$
Answer
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:
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 $i^2 = -1$.
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.
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.
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.
Difference of Squares: This is a pattern that allows us to factor expressions of the form $a^2 - b^2$ into $(a + b)(a - b)$.
Simplifying Complex Expressions: This involves combining like terms, factoring, and reducing expressions to their simplest form.
