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
Solution:
Step 1:
Multiply both the numerator and denominator of $\frac{60i}{3+i}$ by the complex conjugate of the denominator, which is $3-i$. Thus, we have $\frac{60i}{3+i} \cdot \frac{3-i}{3-i}$.
Step 2:
Proceed with the multiplication.
Step 2.1:
Combine terms: $\frac{60i(3-i)}{(3+i)(3-i)}$.
Step 2.2:
Expand the numerator.
Step 2.2.1:
Distribute $60i$ across $(3-i)$: $\frac{60i \cdot 3 - 60i \cdot i}{(3+i)(3-i)}$.
Step 2.2.2:
Calculate $60i \cdot 3$: $\frac{180i - 60i \cdot i}{(3+i)(3-i)}$.
Step 2.2.3:
Multiply $-60i$ by $i$.
Step 2.2.3.1:
Multiply $-60$ by $i$: $\frac{180i - 60i^2}{(3+i)(3-i)}$.
Step 2.2.3.2:
Square $i$: $\frac{180i - 60(i \cdot i)}{(3+i)(3-i)}$.
Step 2.2.3.3:
Combine the powers of $i$: $\frac{180i - 60i^{2}}{(3+i)(3-i)}$.
Step 2.2.3.4:
Apply the exponent rule $i^2 = -1$: $\frac{180i - 60(-1)}{(3+i)(3-i)}$.
Step 2.2.4:
Simplify the numerator.
Step 2.2.4.1:
Substitute $i^2$ with $-1$: $\frac{180i + 60}{(3+i)(3-i)}$.
Step 2.2.4.2:
Reorder terms: $\frac{60 + 180i}{(3+i)(3-i)}$.
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: $\frac{60 + 180i}{3(3-i) + i(3-i)}$.
Step 2.3.1.2:
Multiply out: $\frac{60 + 180i}{9 - 3i + 3i - i^2}$.
Step 2.3.2:
Simplify the denominator.
Step 2.3.2.1:
Combine like terms: $\frac{60 + 180i}{9 - i^2}$.
Step 2.3.2.2:
Substitute $i^2$ with $-1$: $\frac{60 + 180i}{9 + 1}$.
Step 2.3.3:
Add the real numbers in the denominator: $\frac{60 + 180i}{10}$.
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 $10$: $\frac{10(6 + 18i)}{10}$.
Step 3.2:
Cancel the common factor of $10$: $\frac{6 + 18i}{1}$.
Step 3.3:
Simplify the expression: $6 + 18i$.
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 $a + bi$ is $a - bi$. Multiplying a complex number by its conjugate results in a real number because $(a+bi)(a-bi) = a^2 - (bi)^2 = a^2 + b^2$ since $i^2 = -1$.
Distributive Property: This property states that $a(b + c) = ab + ac$. It is used to expand expressions by distributing a single term across terms inside parentheses.
FOIL Method: This stands for First, Outer, Inner, Last. It is a technique for multiplying two binomials. For example, $(a+b)(c+d) = ac + ad + bc + bd$.
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.
