Simplify (60i)/(3+i)

The given problem is a mathematical expression that involves complex numbers. It requires the simplification of a fraction where the numerator is a pure imaginary number (60i) and the denominator is a complex number (3+i). The task is to perform the appropriate mathematical operations to simplify this expression to its simplest form, which should also be expressed as a complex number, involving both a real part and an imaginary part if applicable. Simplification of such expressions typically involves the use of the complex conjugate to eliminate the imaginary unit, i, from the denominator.

$\frac{60 i}{3 + i}$

Answer

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Solution:

Step 1:

Multiply both the numerator and denominator of $\frac{60 i}{3 + i}$ by the complex conjugate of the denominator, which is $3 - i$ . Thus, we have $\frac{60 i}{3 + i} \cdot \frac{3 - i}{3 - i}$ .

Step 2:

Proceed with the multiplication.

Step 2.1:

Combine terms: $\frac{60 i (3 - i)}{(3 + i) (3 - i)}$ .

Step 2.2:

Expand the numerator.

Step 2.2.1:

Distribute $60 i$ across $(3 - i)$ : $\frac{60 i \cdot 3 - 60 i \cdot i}{(3 + i) (3 - i)}$ .

Step 2.2.2:

Calculate $60 i \cdot 3$ : $\frac{180 i - 60 i \cdot i}{(3 + i) (3 - i)}$ .

Step 2.2.3:

Multiply $- 60 i$ by $i$ .

Step 2.2.3.1:

Multiply $- 60$ by $i$ : $\frac{180 i - 60 i^{2}}{(3 + i) (3 - i)}$ .

Step 2.2.3.2:

Square $i$ : $\frac{180 i - 60 (i \cdot i)}{(3 + i) (3 - i)}$ .

Step 2.2.3.3:

Combine the powers of $i$ : $\frac{180 i - 60 i^{2}}{(3 + i) (3 - i)}$ .

Step 2.2.3.4:

Apply the exponent rule $i^{2} = - 1$ : $\frac{180 i - 60 (- 1)}{(3 + i) (3 - i)}$ .

Step 2.2.4:

Simplify the numerator.

Step 2.2.4.1:

Substitute $i^{2}$ with $- 1$ : $\frac{180 i + 60}{(3 + i) (3 - i)}$ .

Step 2.2.4.2:

Reorder terms: $\frac{60 + 180 i}{(3 + i) (3 - i)}$ .

Step 2.3:

Expand the denominator.

Step 2.3.1:

Use the FOIL method to multiply $(3 + i) (3 - i)$ .

Step 2.3.1.1:

Distribute: $\frac{60 + 180 i}{3 (3 - i) + i (3 - i)}$ .

Step 2.3.1.2:

Multiply out: $\frac{60 + 180 i}{9 - 3 i + 3 i - i^{2}}$ .

Step 2.3.2:

Simplify the denominator.

Step 2.3.2.1:

Combine like terms: $\frac{60 + 180 i}{9 - i^{2}}$ .

Step 2.3.2.2:

Substitute $i^{2}$ with $- 1$ : $\frac{60 + 180 i}{9 + 1}$ .

Step 2.3.3:

Add the real numbers in the denominator: $\frac{60 + 180 i}{10}$ .

Step 3:

Reduce the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is $10$ .

Step 3.1:

Factor out $10$ : $\frac{10 (6 + 18 i)}{10}$ .

Step 3.2:

Cancel the common factor of $10$ : $\frac{6 + 18 i}{1}$ .

Step 3.3:

Simplify the expression: $6 + 18 i$ .

Knowledge Notes:

The problem involves simplifying a complex fraction. The key steps in solving this problem include:

Complex Conjugate: The complex conjugate of a number $a + b i$ is $a - b i$ . Multiplying a complex number by its conjugate results in a real number because $(a + b i) (a - b i) = a^{2} - (b i)^{2} = a^{2} + b^{2}$ since $i^{2} = - 1$ .
Distributive Property: This property states that $a (b + c) = a b + a c$ . It is used to expand expressions by distributing a single term across terms inside parentheses.
FOIL Method: This stands for First, Outer, Inner, Last. It is a technique for multiplying two binomials. For example, $(a + b) (c + d) = a c + a d + b c + b d$ .
Simplifying Complex Fractions: To simplify a complex fraction, you can multiply the numerator and the denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator.
Reducing Fractions: To reduce a fraction, divide both the numerator and the denominator by their greatest common divisor.
Properties of $i$ : The imaginary unit $i$ is defined such that $i^{2} = - 1$ . Higher powers of $i$ follow a pattern: $i^{3} = i^{2} \cdot i = - i$ , $i^{4} = i^{2} \cdot i^{2} = 1$ , and so on.

By applying these concepts, the complex fraction is simplified to a standard form where the denominator is a real number, and the fraction is reduced to its simplest form.