Problem 42 - Algebra and Trigonometry Real Mathematics, Real People

Question

Perform the operation and write the result in standard form. $$(\sqrt{3}+\sqrt{15} i)(\sqrt{3}-\sqrt{15} i)$$

Step-by-Step Solution

Answer

The result of \( (\sqrt{3}+\sqrt{15} i)(\sqrt{3}-\sqrt{15} i) \) in standard form is 18 + 0i.

Step 1: Identify the numbers

In the problem, two complex numbers are provided: \(\sqrt{3}+\sqrt{15} i\) and \(\sqrt{3}-\sqrt{15} i\) to be multiplied.

Step 2: Apply the property of Complex numbers

Using the property of multiplication \((a+b)(a-b)= a^2 - b^2\), we obtain \[(\sqrt{3})^2 - (\sqrt{15}i)^2\].

Step 3: Calculate

Now, solving \[(\sqrt{3})^2 - (\sqrt{15}i)^2\] we get 3 - (-15) (since \(i^2 = -1\)). So, the expression simplifies to 3 + 15.

Step 4: Simplify and Write in Standard Form

Adding 3 + 15 gives us 18. So the result in standard form is 18 + 0i.

Key Concepts

Complex Conjugates

Complex conjugates are very important when working with complex numbers. In a complex number, like \(a + bi\), the complex conjugate is \(a - bi\). It's like flipping the sign of the imaginary part.
Conjugates are useful because they help in simplifying expressions, especially when multiplying complex numbers or when finding the magnitude. In the original exercise, the numbers \(\sqrt{3}+\sqrt{15} i\) and \(\sqrt{3}-\sqrt{15} i\) are complex conjugates.
Here’s the interesting part: when you multiply a complex number by its conjugate, the result is a real number. This is because the imaginary parts cancel each other out. This is a crucial point, so remember that this technique often simplifies problems significantly, making your calculations easier.
When practicing, try to identify complex conjugates in various exercises. It will help you get more comfortable with these types of numbers.

Multiplication of Complex Numbers

Multiplying complex numbers can seem tricky at first, but it's easier when you apply some simple algebraic rules. With complex numbers, you have an imaginary part, \(bi\), and a real part, \(a\). Combining these gives us a complex number \(a+bi\).
When multiplying two complex numbers, say \((a+bi)(c+di)\), think of it like multiplying two binomials. You use the distributive property—often referred to as FOIL (First, Outer, Inner, Last)—to multiply each part:

• First: Multiply the first terms: \(a \cdot c\)
• Outer: Multiply the outer terms: \(a \cdot di\)
• Inner: Multiply the inner terms: \(bi \cdot c\)
• Last: Multiply the last terms: \(bi \cdot di\)

While working through this, remember that \(i^2\) is \(-1\). This fact will simplify the expression to get a real number plus an imaginary part. In our exercise, we simply used the formula \((a+b)(a-b)= a^2 - b^2\) to multiply because the terms were conjugates, allowing for an even quicker simplification.

Standard Form of Complex Numbers

The standard form of complex numbers is important as it is the way we usually express these numbers. It’s written as \(a + bi\), where \(a\) is the real part and \(b\) is the coefficient of the imaginary part \(i\).
In our exercise, the final result was 18, which in standard form is expressed as \(18 + 0i\). Writing it like this clearly shows that the imaginary part is zero.
Most problems involving complex numbers will ask you to express your final answer in its standard form. This ensures clarity, especially in multi-step problems where keeping track of real and imaginary parts is crucial. Always remember:

• a: Represents the real component
• b: Represents the coefficient of the imaginary component

So whenever you solve any problem involving complex numbers, checking that your final answer is in standard form can help you verify the correctness. Practice putting expressions into this form to become familiar with it!