Simplify ( square root of 7+5i)/( square root of 5+2i)
The given problem is asking to perform a complex division operation where both the numerator and denominator contain complex numbers under square root signs. The goal is to simplify the expression ( √(7+5i) ) / ( √(5+2i) ) to a standard form, which typically involves rationalizing the denominator to eliminate any complex numbers from it, leading to a result where both the real and imaginary parts are clearly identified, and no radicals appear in the denominator.
$\frac{\sqrt{7} + 5 i}{\sqrt{5} + 2 i}$
Answer
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:
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 $i^2 = -1$.
Conjugate: The conjugate of a complex number $a + bi$ is $a - bi$. Multiplying a complex number by its conjugate results in a real number.
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.
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.
Distributive Property: This property states that $a(b + c) = ab + ac$. It allows the multiplication of a term across the terms within parentheses.
Difference of Squares: This is a pattern that emerges when expanding $(a + b)(a - b)$, resulting in $a^2 - b^2$.
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 $i^2 = -1$.
