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Chapter 10: Problem 49

Add or subtract as indicated. Give answers in standard form. $$ [(7+3 i)-(4-2 i)]+(3+i) $$

Short Answer

Expert verified
6 + 6i.

Step by step solution

01

- Simplify inside the first set of parentheses

To simplify odiscard{C++}[...implies]{(7+3i)-(4-2i)}, distribute the negative sign across the second term: `(7+3i) - 4 + 2i`.
02

- Combine the real and imaginary parts

Combine the real parts and the imaginary parts separately: odiscard{C++}[code-sample2]{(7-4)+(3i+2i) = 3 + 5i}.
03

- Add the result to the remaining term

Add the result from Step 2 to odiscard{C++}[example-sentence]{(3+i)}, combining the real and the imaginary parts: (3 + 5i) + (3 + i).
04

- Combine all real and all imaginary parts

Combine the real parts and the imaginary parts separately again: (3+3) + (5i+i) = 6 + 6i.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combining Like Terms
When working with complex numbers, one of the key skills is to combine like terms. Like terms are terms that contain the same variable or, in the case of complex numbers, the same parts: the real numbers and the imaginary numbers. For example, if you have the expression \(7 + 3i - 4 + 2i\), you can combine the real parts (7 and -4) and the imaginary parts (3i and 2i) separately. This helps in simplification. So, combining 7 and -4 gives 3, and adding 3i and 2i gives 5i. This simplifies the expression to 3 + 5i.
Imaginary Numbers
Imaginary numbers are a fundamental part of complex numbers. They are based on the imaginary unit \i\, which is defined as the square root of -1. This means \(i^2 = -1\). Imaginary numbers are used in a variety of mathematical contexts. When adding or subtracting complex numbers, you need to keep track of the imaginary parts separately from the real numbers. In our example, the imaginary parts were 3i, 2i, and i. When simplifying, we added these together separately, just like we do with the real parts.
Distributive Property
The distributive property is a handy tool when simplifying expressions, including those with complex numbers. It allows us to multiply a single term by every term inside a parenthesis. For instance, in the expression \(7 + 3i - (4 - 2i)\), the distributive property tells us to distribute the negative sign across both terms inside the parentheses: \(7 + 3i - 4 + 2i\). This helps break down the problem into simpler parts. When using the distributive property, it's important to apply it consistently to avoid mistakes.

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Most popular questions from this chapter

Simplify. Assume that all variables represent positive real numbers. \(\sqrt{72 k^{2}}\)

The windchill factor is a measure of the cooling effect that the wind has on a person's skin. It calculates the equivalent cooling temperature if there were no wind. The National Weather Service uses the formula $$ \text { Windchill temperature }=35.74+0.6215 T-35.75 V^{4 / 25}+0.4275 T V^{4 / 25} $$ where T is the temperature in \(^{\circ} \mathrm{F}\) and \(\mathrm{Vi}\) is the wind speed in miles per hour, to calculate wind- chill. The table gives the windchill factor for various wind speeds and temperatures at which frostbite is a risk, and how quickly it may occur. Use the formula to determine the windchill temperature to the nearest tenth of a degree, given the following conditions. Compare answers with the appropriate entries in the table. \(30^{\circ} \mathrm{F}, 15\) -mph wind

Simplify. Assume that all variables represent positive real numbers. \(\sqrt{\frac{y^{11}}{36}}\)

Work each problem. Suppose someone claims that \(\sqrt[n]{a^{n}+b^{n}}\) must equal \(a+b,\) because when \(a=1\) and \(b=0,\) a true statement results: $$ \sqrt[n]{a^{n}+b^{n}}=\sqrt[n]{1^{n}+0^{n}}=\sqrt[n]{1^{n}}=1=1+0=a+b $$ Explain why this is faulty reasoning.

Work each problem. Replace \(a\) with 3 and \(b\) with 4 to show that, in general, $$ \sqrt{a^{2}+b^{2}} \neq a+b $$
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