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MEGA Mathematics: Practice & Study Guide56 chapters | 545 lessons | 23 flashcard sets

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Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn the process of simplifying square roots in complex expressions. Learn how to use the letter i and how to reduce your square roots.

In this video lesson, we will talk about **complex expressions**. These are the mathematical expressions that include complex numbers, which contain both a real part and an imaginary part. Think of the imaginary part of a mathematical expression as your shadow when you are in the shade. You know you have a shadow, but when you're in the shade, you don't see it. The same is true with a complex expression; you know it has an imaginary part, but you just can't see it on the number line.

Complex numbers look like binomials in that they have two terms. For example, 3 + 4*i* is a complex number as well as a complex expression. It looks like a binomial with its two terms. See the letter *i*? That tells you which part is the imaginary part. In this case, 4 is the imaginary part.

Let's talk about *i* for a bit. We define ** i** as the square root of -1. So,

When it comes to simplifying our complex expressions, the letter *i* comes in handy when we need to deal with negative square roots. Because the *i* is defined as the square root of -1, we can use it to simplify any square root. For example, while the square root of -9 is not valid in the real world, by using the letter *i* we can simplify it to 3*i* in a complex expression.

How did we simplify the square root of -9 to 3*i*? We split the square root of -9 into two square roots multiplied with each other. We have the square root of -1 times the square root of 9. We know that the square root of 9 is 3, and from our definition of *i* we have the square root of -1 as *i*. So, we end up with the square root of -9 being 3*i*.

Let's look at simplifying another square root: the square root of -32. How do we do that? We first split this square root into the square root of -1 times the square root of 32. Then we can substitute an *i* for the square root of -1. Now, we can work on simplifying the square root of 32. We can split the square root of 32 into the square root of 2 times the square root of 16. We can split any square root into any two numbers that when multiplied together gives us our original number. We should know the square root of one of those numbers.

In our case, we chose 2 and 16 because 2 * 16 = 32, and because we know the square root of 16. So, we have the square root of 2 times the square root of 16 equals the square root of 32. We know the square root of 16 is 4, so we can substitute a 4 for the square root of 16. The square root of 2 we can't do anything about so we leave it as that. After simplifying, we have the square root of -32 equals 4*i* times the square root of 2.

Now that we know how to simplify our square roots, we can very easily simplify any complex expression with square roots in it. We use a combination of algebra skills and simplifying square root skills to simplify our complex expressions. We treat *i* as a variable and any numbers attached to it as a coefficient.

Let's look at an example. To simplify 5 + *i* + the square root of -36 we first simplify the square root of -36. To do this, we first split the square root of -36 into the square root of -1 and the square root of 36. The square root of 36 simplifies to 6 and we have the square root of -36 is 6*i*. Our expression is now 5 + *i* + 6*i*. Using our algebra skills, we combine the *i* and the 6*i* to become 7*i*. Our simplified expression is 5 + 7*i*. We are done.

Let's try another one. Simplify 3 + 4 * the square root of -3. We first simplify the square root of -3. We split the -3 into -1 and 3 to get the square root of -1 times the square root of 3. We can't do anything with the square root of 3, but we can substitute an *i* for the square root of -1. Our expression changes to 3 + 4*i* * the square root of 3, and we are done.

What have we learned now? We learned that **complex expressions** are the mathematical expressions that include complex numbers, which contain both a real part and an imaginary part. The imaginary part is represented by a letter *i*.

We define ** i** as the square root of -1 and

There is no such thing as the square root of a negative number in the real world, but in the imaginary world, it is represented with an *i*. For example, the square root of -9 is 3*i*. To simplify any square root we split the square root into two square roots where the two numbers multiply to our original numbers and where we know the square root of one of the numbers.

For example, to simplify the square root of 32 we split it into the square root of 2 and the square root of 16 because 2 * 16 = 32 and we know what the square root of 16 equals. To simplify complex expressions, we use both our algebra skills and our simplifying square roots skills.

After you are finished with this lesson you should be able to:

- Describe a complex expression
- Define the letter
*i* - Simplify negative square roots and complex expressions

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MEGA Mathematics: Practice & Study Guide56 chapters | 545 lessons | 23 flashcard sets

- Go to Decimals

- Go to Percents

- Go to Vectors

- How to Find the Square Root of a Number 5:42
- Estimating Square Roots 5:10
- Simplifying Square Roots When not a Perfect Square 4:45
- Simplifying Complex Expressions That Contain Square Roots 6:43
- Evaluating Square Roots of Perfect Squares 5:12
- Factoring Radical Expressions 4:45
- Simplifying Square Roots of Powers in Radical Expressions 3:51
- Multiplying then Simplifying Radical Expressions 3:57
- Dividing Radical Expressions 7:07
- Simplify Square Roots of Quotients 4:49
- Rationalizing Denominators in Radical Expressions 7:01
- Addition and Subtraction Using Radical Notation 3:08
- Multiplying Radical Expressions with Two or More Terms 6:35
- Solving Radical Equations: Steps and Examples 6:48
- Solving Radical Equations with Two Radical Terms 6:00
- Go to Radical Expressions

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