Simplify ( square root of 7+5i)/( square root of 5+2i)

Solution:

Step:1

Rationalize the denominator by multiplying $\frac{\sqrt{7}+5i}{\sqrt{5}+2i}$ with its conjugate $\frac{\sqrt{5}-2i}{\sqrt{5}-2i}$: $\frac{\sqrt{7}+5i}{\sqrt{5}+2i}\cdot \frac{\sqrt{5}-2i}{\sqrt{5}-2i}$

Step:2

Combine the fractions into a single expression.

Step:2.1

Perform the multiplication of numerators and denominators: $\frac{(\sqrt{7}+5i)(\sqrt{5}-2i)}{(\sqrt{5}+2i)(\sqrt{5}-2i)}$

Step:2.2

Expand the denominator using the difference of squares: $\frac{(\sqrt{7}+5i)(\sqrt{5}-2i)}{(\sqrt{5}{)}^2-(2i{)}^2}$

Step:2.3

Simplify the denominator: $\frac{(\sqrt{7}+5i)(\sqrt{5}-2i)}{5-(-4)}=\frac{(\sqrt{7}+5i)(\sqrt{5}-2i)}{9}$

Step:3

Expand the numerator using the distributive property (FOIL Method).

Step:3.1

Distribute the terms: $\frac{\sqrt{7}(\sqrt{5}-2i)+5i(\sqrt{5}-2i)}{9}$

Step:3.2

Continue with the distribution: $\frac{\sqrt{7}\sqrt{5}-2i\sqrt{7}+5i\sqrt{5}-10i^2}{9}$

Step:4

Simplify the terms in the numerator.

Step:4.1

Simplify each term individually.

Step:4.1.1

Combine the radicals: $\frac{\sqrt{35}-2i\sqrt{7}+5i\sqrt{5}-10i^2}{9}$

Step:4.1.2

No further simplification needed for $\sqrt{35}$.

Step:4.1.3

Simplify the term with $i^2$.

Step:4.1.3.1

No further simplification needed for $-2i\sqrt{7}$.

Step:4.1.3.2

No further simplification needed for $5i\sqrt{5}$.

Step:4.1.3.3

Recognize that $i^2=-1$.

Step:4.1.3.4

Apply the identity $i^2=-1$: $\frac{\sqrt{35}-2i\sqrt{7}+5i\sqrt{5}+10}{9}$

Step:4.1.3.5

Combine the real parts and the imaginary parts: $\frac{10+\sqrt{35}-2i\sqrt{7}+5i\sqrt{5}}{9}$

Step:4.2

The expression is already ordered correctly.

Knowledge Notes:

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

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

3. Rationalizing the Denominator: This process involves eliminating the radical or imaginary number in the denominator by multiplying the numerator and denominator by the conjugate of the denominator.

4. FOIL Method: A technique for expanding the product of two binomials. It stands for First, Outer, Inner, Last, referring to the terms that are multiplied together.

5. Distributive Property: This property states that $a(b+c)=ab+ac$. It allows the multiplication of a term across the terms within parentheses.

6. Difference of Squares: This is a pattern that emerges when expanding $(a+b)(a-b)$, resulting in $a^2-b^2$.

7. Simplifying Complex Expressions: When simplifying expressions involving complex numbers, it's important to separate and simplify the real and imaginary parts, often involving the use of the identity $i^2=-1$.