Simplify (2-i)/5(i+2)
The question asks you to perform a complex number simplification. Specifically, it requires you to simplify the given fraction, which has a complex number, 2-i (where i is the imaginary unit, √(-1)), in the numerator and another complex number, 5(i+2), in the denominator. The goal is to simplify this expression to its standard form, where the result should have a real part and an imaginary part, each properly simplified and expressed in terms of the imaginary unit i. To approach this problem, you would typically multiply both the numerator and the denominator by the conjugate of the denominator to remove the imaginary part from the denominator and simplify further to express the result in terms of a + bi, where a and b are real numbers.
$\frac{2 - i}{5} \left(\right. i + 2 \left.\right)$
Answer
Solution:
Step 1:
Divide the complex fraction $\frac{2 - i}{5(i + 2)}$ into real and imaginary parts: $\left(\frac{2}{5} - \frac{i}{5}\right)(i + 2)$.
Step 2:
Position the negative sign correctly: $\left(\frac{2}{5} - \frac{i}{5}\right)(i + 2)$.
Step 3:
Distribute each part of the first term over the second term.
Step 3.1:
Use the distributive property: $\frac{2}{5}(i + 2) - \frac{i}{5}(i + 2)$.
Step 3.2:
Continue with the distributive property: $\frac{2i}{5} + \frac{4}{5} - \frac{i^2}{5} - \frac{2i}{5}$.
Step 4:
Simplify the expression by combining like terms.
Step 4.1:
Simplify each term individually.
Step 4.1.1:
Combine the terms with $i$: $\frac{2i}{5} + \frac{4}{5} - \frac{i^2}{5} - \frac{2i}{5}$.
Step 4.1.2:
Calculate the product of $\frac{2}{5}$ and $2$: $\frac{2i}{5} + \frac{4}{5} - \frac{i^2}{5} - \frac{2i}{5}$.
Step 4.1.3:
Multiply $-\frac{i}{5}$ by $i$: $\frac{2i}{5} + \frac{4}{5} - \frac{i^2}{5} - \frac{2i}{5}$.
Step 4.1.4:
Substitute $i^2$ with $-1$: $\frac{2i}{5} + \frac{4}{5} + \frac{1}{5} - \frac{2i}{5}$.
Step 4.1.5:
Rearrange the negative sign: $\frac{2i}{5} + \frac{4}{5} + \frac{1}{5} - \frac{2i}{5}$.
Step 4.1.6:
Calculate the product of $-\frac{1}{5}$: $\frac{2i}{5} + \frac{4}{5} + \frac{1}{5} - \frac{2i}{5}$.
Step 4.1.7:
Multiply $-\frac{i}{5}$ by $2$: $\frac{2i}{5} + \frac{4}{5} + \frac{1}{5} - \frac{2i}{5}$.
Step 4.1.8:
Position the negative sign correctly: $\frac{2i}{5} + \frac{4}{5} + \frac{1}{5} - \frac{2i}{5}$.
Step 4.2:
Add the real parts together: $\frac{4}{5} + \frac{1}{5}$.
Step 4.3:
Add the imaginary parts together: $\frac{2i}{5} - \frac{2i}{5}$.
Step 4.4:
Combine the real parts: $\frac{4 + 1}{5}$.
Step 4.5:
Simplify the fraction: $\frac{5}{5}$.
Step 5:
Conclude that the simplified form is $1$.
Knowledge Notes:
To simplify a complex fraction, especially one involving complex numbers, the following knowledge points are relevant:
Complex Numbers: A complex number is of the form $a + bi$, where $a$ is the real part, $b$ is the imaginary part, and $i$ is the imaginary unit with the property that $i^2 = -1$.
Distributive Property: This property states that $a(b + c) = ab + ac$. It is used to expand expressions by distributing a single term over a sum or difference within parentheses.
FOIL Method: This stands for First, Outer, Inner, Last. It is a specific case of the distributive property used for multiplying two binomials.
Combining Like Terms: Terms that have the same variable part can be combined by adding or subtracting their coefficients.
Simplifying Complex Fractions: When simplifying complex fractions, it's often helpful to separate the real and imaginary parts and simplify them separately.
Multiplying Complex Numbers: When multiplying complex numbers, one must consider the product of the real parts, the product of the imaginary parts, and the cross products.
Power Rule for Exponents: For any nonzero number $a$, the power rule $a^m a^n = a^{m+n}$ is used to combine like bases with exponents by adding the exponents.
By applying these principles, one can systematically simplify complex fractions involving complex numbers.
