Question

Multiply. Write the answer in standard form. \((4-2 i)(4+2 i)\)

Short answer

The simplified standard form of the given complex expression is 20

Step-by-step solution

Distribute

Distribute (or foil) the terms in the expressions: \(4*4 + 4*2i - 2i*4 - 2i*2i\) which equals \(16 + 8i - 8i - 4i^2\)

Simplify

Combine like terms and use the fact that \(i^2 = -1\). The combined equation is thus: \(16 - 4*(-1)\)

Final Calculation

Now, perform the final calculation: \(16+4 = 20\)

Key concepts

Multiplying Complex Numbers

When multiplying complex numbers, it's important to use the distributive property to ensure each term is correctly multiplied. You can think of this process as similar to multiplying binomials. It involves using the FOIL method, which stands for First, Outer, Inner, and Last. This mnemonic helps you keep track of the terms you're combining from each part of the numbers.

• **First**: Multiply the first terms in each pair.
• **Outer**: Multiply the outer terms in each pair.
• **Inner**: Multiply the inside terms in each pair.
• **Last**: Multiply the last terms in each pair.

By carefully applying this method, you can confidently multiply complex numbers to find the correct product. After applying these steps, the complex parts will often cancel each other out or combine, simplifying the final answer.

Standard Form in Algebra

In algebra, standard form is a way of writing down the solution in a neat, ordered way. For complex numbers, standard form is written as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. This format is important because it clearly separates the real and imaginary components.
When you multiply complex numbers and get to the final step, you need to ensure that your answer is in this form. This involves performing any final calculations and simplifying to eliminate any unnecessary terms.
For instance, if you have a term involving \(i^2\), remember that \(i^2 = -1\). By substituting and simplifying, you can more easily express your answer in standard form, which is neat, organized, and immediately useful for further mathematical operations.

Complex Conjugates

Complex conjugates are special pairs of complex numbers. The conjugate of a complex number \(a + bi\) is \(a - bi\).
These pairs have a unique property: when multiplied together, the imaginary parts cancel out, resulting in a real number.
This is because the product of \(bi\) and \(-bi\) is \(-b^2i^2\), which equals \(b^2\) since \(i^2 = -1\).

• The real part is calculated as \(a^2 + b^2\).
• The imaginay part cancels out, leaving no \(i\) term.

This feature is especially useful for simplifying complex number fractions, as it turns the denominator into a real number, greatly simplifying calculations. Using conjugates is a powerful tool in algebra.