# Complex Numbers - Practice Problems

TLDRThis video tutorial covers various complex mathematical problems involving imaginary numbers and their properties. It explains the calculation of absolute values, simplification of expressions, rationalizing denominators, and solving equations with imaginary units. The detailed step-by-step explanations use methods like distribution, FOIL, and conjugates, ensuring a thorough understanding of each problem. Viewers will learn how to simplify expressions, compute absolute values, and handle imaginary numbers, making it an essential guide for mastering these concepts.

###### Takeaways

- π The script is a detailed walkthrough of complex mathematical problems involving complex numbers and absolute values.
- π’ The first problem explains how to find the absolute value of a complex number, demonstrating the calculation with the example of 6 + 8i.
- π In problem two, the script simplifies a given expression by distributing and combining like terms, leading to the answer choice E.
- π The third problem involves using the FOIL method to simplify a complex expression, resulting in the answer A.
- π For problem four, the script expands and simplifies an expression by squaring a binomial with complex numbers, concluding with answer choice A.
- π The fifth problem explores the powers of the imaginary unit 'i', breaking down \( i^{59} \) into \( i^{4 \cdot 14} \cdot i^3 \) and simplifying to \( -i \).
- π± In the sixth problem, the script simplifies a radical expression involving complex numbers, correctly identifying the final answer as \( \sqrt{6} \).
- π Problem seven focuses on rationalizing the denominator of a complex fraction, leading to a simplified form involving real and imaginary parts.
- π The eighth problem calculates the absolute value of a complex expression, simplifying powers of 'i' and using the Pythagorean theorem to find the magnitude.
- π In problem nine, the script simplifies a complex fraction by multiplying numerator and denominator by the conjugate of the denominator.
- π The final problem, number ten, involves separating real and imaginary parts of an equation to solve for x and y, and then finding their sum.

###### Q & A

### What is the absolute value of the complex number 6 + 8i?

-The absolute value of the complex number 6 + 8i is 10. It is calculated by taking the square root of the sum of the squares of the real and imaginary parts, which is β(6^2 + 8^2) = β(36 + 64) = β100 = 10.

### How do you simplify the expression 5(2 - 3i) - 4(4 + 6i)?

-The expression simplifies to -6 - 39i. This is done by distributing the multiplication: 5*2 - 5*3i - 4*4 - 4*6i, which gives 10 - 15i - 16 - 24i, and then combining like terms: -6 - 39i.

### What is the result of the expression (5 - 2i)(2 + 5i)?

-The result is 29. This is found by using the FOIL method (First, Outer, Inner, Last): 5*2 + 5*5i - 2i*2 - 2i*5i, which simplifies to 10 + 25i - 4i - 10i^2. Since i^2 = -1, this becomes 10 + 21i + 10, which is 29.

### How do you expand and simplify (3 + 7i)^2?

-The expression simplifies to -40 + 42i. This is done by multiplying (3 + 7i) by itself: 3*3 + 2*(3*7i) + (7i)^2, which gives 9 + 42i - 49. Since i^2 = -1, -49 becomes -49*(-1) = 49, and the expression simplifies to 9 - 49 + 42i, which is -40 + 42i.

### What is i raised to the power of 59?

-i^59 is equal to -i. This is because i^59 can be broken down into (i^4)^14 * i^3, and since i^4 = 1, this simplifies to 1^14 * i^3 = i^3 = -i.

### How do you simplify the expression β(-14) as a complex number?

-The expression simplifies to β6 * i. This is because β(-14) can be written as β(7*2) * β(-1), which is β7 * β2 * i, and since β(-1) is i, the expression simplifies to β6 * i.

### What is the process to rationalize the denominator of a complex fraction?

-To rationalize the denominator, you multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate changes the sign of the imaginary part.

### What is the absolute value of 5i^12 - 3i^19?

-The absolute value is β34. This is found by simplifying the expression to 5 + 3i and then taking the square root of the sum of the squares of the real and imaginary parts: β(5^2 + 3^2) = β(25 + 9) = β34.

### How do you simplify the expression (3 + 2i)/(4 + 3i) by multiplying by the conjugate?

-The simplified expression is 6/25 + 17/25i. This is done by multiplying the numerator and the denominator by the conjugate of the denominator (4 - 3i), then simplifying the resulting expression.

### What is the value of x + y in the given complex equation?

-The value of x + y is 1. This is found by separating the real and imaginary parts of the equation and solving for x and y individually, then adding them together: x = 3 and y = -2, so x + y = 3 - 2 = 1.

###### Outlines

##### π Complex Numbers and Absolute Value Calculations

This paragraph discusses the concept of absolute value in the context of complex numbers. It begins with a problem involving the absolute value of 6 + 8i, explaining the process of calculating the magnitude of a complex number by taking the square root of the sum of the squares of its real and imaginary parts. It then moves on to simplify complex expressions involving multiplication and distribution, with examples provided. The paragraph also covers the simplification of powers of 'i', the imaginary unit, and the use of the conjugate to rationalize denominators in complex fractions.

##### π§© Simplifying Complex Expressions and Rationalizing Denominators

The second paragraph continues the theme of complex numbers, focusing on simplifying expressions and rationalizing denominators. It starts by breaking down the square root of a negative number into its components and simplifying it using the properties of 'i'. The paragraph then demonstrates how to rationalize the denominator of a complex fraction by multiplying the numerator and denominator by the conjugate of the denominator. It also touches on finding the absolute value of complex numbers raised to high powers by simplifying them to a form that allows for easy calculation of the magnitude.

##### π Solving for Variables in Complex Equations

In the final paragraph, the focus shifts to solving for variables within complex equations. The process involves separating real and imaginary components to form two distinct equations. By solving these equations individually, the values of the variables x and y are determined. The paragraph illustrates how to isolate the real and imaginary parts, solve for each variable, and then combine these values to find the sum of x and y. The solution process is methodical, emphasizing the importance of correctly identifying and manipulating real and imaginary components.

###### Mindmap

###### Keywords

##### π‘Absolute Value

##### π‘Complex Numbers

##### π‘Conjugate

##### π‘Foil Method

##### π‘Imaginary Unit

##### π‘Rationalize

##### π‘Simplify

##### π‘Exponentiation

##### π‘Roots

##### π‘Equation

###### Highlights

Absolute value of complex numbers is calculated using the square root of the sum of the squares of their real and imaginary parts.

The absolute value of 6 + 8i is found to be 10, demonstrating the use of the Pythagorean theorem in complex numbers.

Simplification of complex expressions involves distributing and combining like terms.

The answer to the first simplification problem is revealed to be choice E, showcasing a step-by-step solution.

FOIL method is applied to simplify expressions involving complex numbers.

Understanding that i squared equals -1 is crucial for simplifying complex expressions.

The solution to the third problem is 29, highlighting the process of simplifying complex multiplication.

Expanding and simplifying expressions with complex numbers involves multiplying and combining like terms.

The absolute value of a complex number can be found by converting it to a + bi format and then taking the square root of the sum of the squares of a and b.

Rationalizing the denominator of a complex fraction involves multiplying by the conjugate.

The conjugate of a complex number is used to eliminate the imaginary part in the denominator.

The process of rationalizing the denominator is demonstrated with a step-by-step approach.

Finding the absolute value of complex expressions raised to powers requires converting to a + bi format first.

The absolute value of a complex number raised to a power is the square root of the sum of the squares of its real and imaginary parts.

Simplifying complex expressions involves multiplying by the conjugate of the denominator to eliminate the imaginary part.

The final answer to the simplification problem is presented in the form of a fraction with real and imaginary parts.

Solving for x and y in a complex equation involves separating real and imaginary parts and solving the resulting equations.

The value of x plus y is found to be 1 by solving the real and imaginary parts of the equation separately.

###### Transcripts

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