Math Problem Statement

Simplify \( \frac{5i}{(2 + 3i)^2} \)

Solution

The expression provided is:

5i(2+3i)2\frac{5i}{(2 + 3i)^2}

To simplify this, we can proceed by expanding the denominator and then rationalizing the fraction.

Step 1: Expand the Denominator

The denominator is (2+3i)2(2 + 3i)^2. Using the formula (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2, we have:

(2+3i)2=22+223i+(3i)2(2 + 3i)^2 = 2^2 + 2 \cdot 2 \cdot 3i + (3i)^2

Calculating each term:

  • 22=42^2 = 4
  • 223i=12i2 \cdot 2 \cdot 3i = 12i
  • (3i)2=9i2=9(1)=9(3i)^2 = 9i^2 = 9 \cdot (-1) = -9

So, the denominator becomes: 4+12i9=5+12i4 + 12i - 9 = -5 + 12i

Step 2: Write the Expression

Now, we have: 5i5+12i\frac{5i}{-5 + 12i}

Step 3: Rationalize the Denominator

To eliminate the imaginary unit from the denominator, we multiply the numerator and the denominator by the complex conjugate of the denominator, which is 512i-5 - 12i:

5i(512i)(5+12i)(512i)\frac{5i \cdot (-5 - 12i)}{(-5 + 12i)(-5 - 12i)}

Expanding the denominator: (5+12i)(512i)=(5)2(12i)2=25(144)=25+144=169(-5 + 12i)(-5 - 12i) = (-5)^2 - (12i)^2 = 25 - (-144) = 25 + 144 = 169

Now, expand the numerator: 5i(512i)=25i60i25i \cdot (-5 - 12i) = -25i - 60i^2

Since i2=1i^2 = -1: 25i60(1)=25i+60=6025i-25i - 60(-1) = -25i + 60 = 60 - 25i

Step 4: Simplify the Result

Now we have: 6025i169\frac{60 - 25i}{169}

This simplifies to: 6016925i169\frac{60}{169} - \frac{25i}{169}

So the final answer is: 6016925i169\frac{60}{169} - \frac{25i}{169}

Would you like a further breakdown of any part of this solution, or do you have any questions?

Here are some related questions to deepen your understanding:

  1. How do you find the modulus of a complex number?
  2. What is the importance of rationalizing the denominator in complex fractions?
  3. How do you multiply complex numbers using their conjugates?
  4. What is the significance of i2=1i^2 = -1 in complex number calculations?
  5. Can you apply this method to a different complex expression?

Tip: When simplifying expressions with complex numbers, always look for opportunities to use conjugates to remove imaginary components from the denominator.

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Math Problem Analysis

Mathematical Concepts

Complex Numbers
Rationalization
Imaginary Numbers

Formulas

Square of a binomial formula: (a + b)^2 = a^2 + 2ab + b^2
Multiplying by the complex conjugate to rationalize the denominator

Theorems

Complex Conjugate Multiplication

Suitable Grade Level

Grades 11-12