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The modulus of a complex number gives its distance from the origin on the complex plane. It's like finding the length of the hypotenuse in a right-angled triangle. If your complex number is in the form of \(a + bi\), the modulus is calculated using the formula: \(\lVert a + bi \rVert = \sqrt{a^2 + b^2}\). This formula comes from the Pythagorean theorem.
In the given problem, we first find the modulus of \(9 + 4i\). Plugging in the values, we get:

• \(a = 9\)
• \(b = 4\)

Now, using the formula: \(\lVert 9 + 4i \rVert = \sqrt{9^2 + 4^2} = \sqrt{81 + 16} = \sqrt{97}\). Similarly, for \(\frac{1 - 7i}{10}\), calculate:

• \(a = 1/10\)
• \(b = -7/10\)

Thus, \(\lVert \frac{1 - 7i}{10} \rVert = \frac{1}{10}\sqrt{1^2 + (-7)^2} = \frac{1}{10}\sqrt{1 + 49} = \frac{\sqrt{50}}{10} = \frac{5\sqrt{2}}{10} = \frac{\sqrt{2}}{2}\). By practicing these steps, you can become proficient in finding the modulus of any complex number.

Complex numbers involve operations similar to real numbers with some additional rules for the imaginary part. Here are the key operations:

• Addition: Combine like terms. For example, \((3 - i) + (1 + 2i) = 4 + i\).
• Subtraction: Distribute the negative sign and combine like terms. For example, \( (3 - i) - (1 + 2i) = 2 - 3i\).
• Multiplication: Use the distributive property and remember that \(i^2 = -1\). For example, \((3 - i)(1 + 2i) = 3 + 6i - i - 2i^2\) which simplifies to \(5 + 5i\).
• Division: Multiply the numerator and the denominator by the conjugate of the denominator. For example, to simplify \(\frac{3 - i}{2 + 4i}\), multiply by the conjugate of the denominator: \(\frac{(3 - i)(2 - 4i)}{(2 + 4i)(2 - 4i)}\). The denominator then becomes a real number.

By consistently practicing these operations, you will find it easier to handle complex numbers in various mathematical contexts.

Mathematical problem-solving often involves breaking down a problem into smaller, more manageable parts. Here’s a step-by-step approach:

• Understand the problem: Carefully read the problem statement. Identify the given information and what needs to be found.
• Devise a plan: Consider different strategies to solve the problem. This could involve algebraic manipulations, geometric interpretations, or other mathematical techniques.
• Execute the plan: Carry out the steps you have planned methodically. Make sure to work step by step to avoid errors.
• Review your work: Once you have an answer, review your calculations and reasoning. Verify if your solution makes sense in the context of the problem.

For example, in the given exercise, the problem is divided into clear steps:
1. Calculate expressions (like \(2a + 3b\) and \( \frac{a}{2b} \)).
2. Simplify the expressions using appropriate operations.
3. Find the modulus of the resulting complex numbers. Breaking problems into these steps simplifies complex calculations and enhances problem-solving efficiency.