Simplify 4/(8-i)

Solution:

Step 1:

Multiply both the numerator and denominator of $\frac{4}{8 - i}$ by the complex conjugate of the denominator, which is $8 + i$. This results in $\frac{4}{8 - i} \cdot \frac{8 + i}{8 + i}$.

Step 2:

Perform the multiplication.

Step 2.1:

Combine terms to form a single fraction: $\frac{4(8 + i)}{(8 - i)(8 + i)}$.

Step 2.2:

Simplify the numerator.

Step 2.2.1:

Use the distributive property to expand: $\frac{4 \cdot 8 + 4 \cdot i}{(8 - i)(8 + i)}$.

Step 2.2.2:

Multiply 4 by 8: $\frac{32 + 4i}{(8 - i)(8 + i)}$.

Step 2.3:

Simplify the denominator.

Step 2.3.1:

Expand $(8 - i)(8 + i)$ using the FOIL method.

Step 2.3.1.1:

Distribute the terms: $\frac{32 + 4i}{8(8 + i) - i(8 + i)}$.

Step 2.3.1.2:

Continue distributing: $\frac{32 + 4i}{8 \cdot 8 + 8 \cdot i - i \cdot 8 - i \cdot i}$.

Step 2.3.2:

Simplify the expression.

Step 2.3.2.1:

Multiply 8 by 8: $\frac{32 + 4i}{64 + 8i - 8i - ii}$.

Step 2.3.2.2:

Multiply -1 by 8: $\frac{32 + 4i}{64 + 8i - 8i - i^2}$.

Step 2.3.2.3:

Recognize that $i$ raised to the power of 1 is $i$: $\frac{32 + 4i}{64 + 8i - 8i - (i^1 \cdot i)}$.

Step 2.3.2.4:

Apply the power rule for exponents: $\frac{32 + 4i}{64 + 8i - 8i - i^{1 + 1}}$.

Step 2.3.2.5:

Add the exponents of $i$: $\frac{32 + 4i}{64 + 8i - 8i - i^2}$.

Step 2.3.2.6:

Simplify $i^2$ to $-1$: $\frac{32 + 4i}{64 - i^2}$.

Step 2.3.2.7:

Combine like terms: $\frac{32 + 4i}{64 + 1}$.

Step 2.3.3:

Simplify the denominator.

Step 2.3.3.1:

Recognize that $i^2 = -1$: $\frac{32 + 4i}{64 - (-1)}$.

Step 2.3.3.2:

Simplify the expression: $\frac{32 + 4i}{65}$.

Step 2.3.4:

Add 64 and 1: $\frac{32 + 4i}{65}$.

Step 3:

Separate the real and imaginary parts of the fraction: $\frac{32}{65} + \frac{4i}{65}$.

Knowledge Notes:

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

2. Distributive Property: This property states that $a(b + c) = ab + ac$. It is used to expand expressions and to simplify the multiplication of binomials.

3. FOIL Method: This is a technique for multiplying two binomials. It stands for First, Outer, Inner, Last, referring to the terms that are multiplied together.

4. Simplifying Complex Fractions: To simplify a fraction with a complex number in the denominator, multiply the numerator and denominator by the complex conjugate of the denominator.

5. Imaginary Unit $i$: The imaginary unit $i$ is defined as $\sqrt{-1}$. Its powers cycle through four different values: $i^0 = 1$, $i^1 = i$, $i^2 = -1$, $i^3 = -i$, and then it repeats.

6. Power Rule for Exponents: For any number $a$, the power rule states that $a^m \cdot a^n = a^{m+n}$.

7. Simplifying Expressions: Combining like terms and using arithmetic operations to reduce expressions to their simplest form.