5 Simple Steps To Simplify Complex Expressions Like 3-4i/3 4i 3 4i/3-4i

Understanding how to simplify complex expressions might seem challenging at first, but it can become straightforward with a bit of guidance. The expression you've provided, ( \frac{3-4i}{3} + 4i - \frac{4i}{3-4i} ), can be tackled through a series of simple steps. In this guide, I'll walk you through five straightforward steps to simplify this complex expression effectively. So, let's dive right in!

Step 1: Break Down the Expression

First things first! Let's dissect the expression into manageable parts:

[ \frac{3-4i}{3} + 4i - \frac{4i}{3-4i} ]

This expression contains three different terms: a fraction ( \frac{3-4i}{3} ), a whole number with an imaginary component ( 4i ), and another fraction ( -\frac{4i}{3-4i} ).

Step 2: Simplify Each Term Individually

Simplifying ( \frac{3-4i}{3} )

For the first term, simply divide both parts of the numerator by the denominator:

[ \frac{3}{3} - \frac{4i}{3} = 1 - \frac{4i}{3} ]

Simplifying ( -\frac{4i}{3-4i} )

Next, let's simplify the fraction ( -\frac{4i}{3-4i} ). To do this, we can multiply the numerator and denominator by the conjugate of the denominator, which is ( 3 + 4i ):

[ -\frac{4i(3 + 4i)}{(3 - 4i)(3 + 4i)} = -\frac{12i + 16i^2}{9 + 16} = -\frac{12i - 16}{25} ]

This reduces to:

[ \frac{16 - 12i}{25} ]

Summary of Simplified Terms

Now, we have simplified each term:

1. ( \frac{3-4i}{3} = 1 - \frac{4i}{3} )
2. ( 4i ) remains as is.
3. ( -\frac{4i}{3-4i} = \frac{16 - 12i}{25} )

Step 3: Combine All Terms Together

Now that we've simplified each term, let's combine them back into a single expression:

[ 1 - \frac{4i}{3} + 4i + \frac{16 - 12i}{25} ]

Finding a Common Denominator

To combine the terms efficiently, we should look for a common denominator. The least common multiple of ( 3 ) and ( 25 ) is ( 75 ).

Rewriting each term with the common denominator:

1. First term: ( 1 = \frac{75}{75} )
2. Second term: ( -\frac{4i}{3} = -\frac{100i}{75} )
3. Third term: ( 4i = \frac{300i}{75} )
4. Fourth term: ( \frac{16 - 12i}{25} = \frac{3(16 - 12i)}{75} = \frac{48 - 36i}{75} )

Now, combine:

[ \frac{75 - 100i + 300i + 48 - 36i}{75} ]

Step 4: Combine Like Terms

Now we can combine like terms in the numerator:

Real Part:

[ 75 + 48 = 123 ]

Imaginary Part:

[ -100i + 300i - 36i = 164i ]

So the expression becomes:

[ \frac{123 + 164i}{75} ]

Step 5: Final Simplified Form

Thus, the final simplified expression is:

[ \frac{123}{75} + \frac{164i}{75} ]

This can further be simplified (if desired) to:

[ \frac{41}{25} + \frac{164i}{75} ]

This process might feel tedious, but with practice, simplifying complex expressions will become easier and more intuitive! Let’s take a moment to highlight some helpful tips, common mistakes, and answers to questions you might have.

Common Mistakes to Avoid

• Miscalculating the conjugate: Remember that when multiplying by the conjugate, both terms need to be flipped in sign.
• Forgetting to combine like terms: Always ensure to add or subtract both the real and imaginary parts separately.
• Overlooking simplification: Check if any fractions can be reduced or if terms can combine for a cleaner final answer.

Troubleshooting Tips

If you find yourself stuck:

• Break the expression down into smaller parts as we've done.
• Double-check your calculations at each step.
• Use a calculator if needed to verify complex arithmetic.

<div class="faq-section"> <div class="faq-container"> <h2>Frequently Asked Questions</h2> <div class="faq-item"> <div class="faq-question"> <h3>What is a complex expression?</h3> <span class="faq-toggle">+</span> </div> <div class="faq-answer"> <p>A complex expression involves real and imaginary numbers, typically represented in the form a + bi, where a and b are real numbers, and i is the imaginary unit.</p> </div> </div> <div class="faq-item"> <div class="faq-question"> <h3>Why do I need to simplify complex expressions?</h3> <span class="faq-toggle">+</span> </div> <div class="faq-answer"> <p>Simplifying complex expressions helps to make calculations easier and enables clearer understanding of the components involved in the expression.</p> </div> </div> <div class="faq-item"> <div class="faq-question"> <h3>How do I multiply complex numbers?</h3> <span class="faq-toggle">+</span> </div> <div class="faq-answer"> <p>To multiply complex numbers, apply the distributive property (FOIL) and combine like terms while remembering that i² = -1.</p> </div> </div> </div> </div>

Recapping what we've discussed, we broke down and simplified a complex expression step by step. We learned the significance of combining like terms, identifying common denominators, and avoiding common mistakes. It’s essential to practice these techniques to master simplifying complex expressions.

Now, I encourage you to practice simplifying similar expressions and explore more tutorials on this topic to strengthen your skills! Happy simplifying!

<p class="pro-note">🌟Pro Tip: Always visualize complex numbers in a plane to better understand their behavior!</p>