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Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

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Lesson Transcript

Instructor:
*Jeff Calareso*

Jeff teaches high school English, math and other subjects. He has a master's degree in writing and literature.

What is a square root? In this lesson, we'll learn what a square root is and how to find it. We'll review a variety of examples in order to master the concept.

Did you ever grow a plant in a glass so you could watch its roots grow? Did you ever see roots grow into a perfect square? No? Me neither. And, in fact, plant roots growing into polygons have nothing to do with square roots.

So what are square roots? Before we can learn how to find square roots, let's quickly review squares. The **square** of a number is just a number multiplied by itself. If our number is *n*, then *n* to the second power is *n* squared. We'd write 3 squared like this:

The square just means it's raised to a power of two, or has an exponent of two, which means the same thing.

So why is it called a square? Imagine you're planting seeds in the actual ground - no glasses required. Let's say it's tomatoes. You want to form a square of tomato plants that's 3 by 3. That's one, two, three along the bottom below and one, two, three along the side.

How many plants will you have? 3 * 3, or 9. That's why a number multiplied by itself is called a square.

Some numbers are called perfect squares. 9 is one example. That means it's a number that you'd get by squaring a whole number. Start with 1. 1 is a perfect square; it's 1 * 1. Then 4, which is 2 * 2, then 9, then 16, 25, 36, 49, 64, 81 and 100. 100 is just 10 * 10. Perfect squares just keep going from there.

That leads us to square roots. A **square root** is the inverse of squaring a number. In other words, the square root of *n*^2 is *n*.

We use this symbol to indicate a square root:

This is called a radical sign. Radical isn't just a word used by '80s surfers and Ninja Turtles. In math, the word 'radical' actually means 'root.' And you know what? That line stretching down kind of looks like the root of one of our tomato plants diving into the ground.

This symbol doesn't always mean the square root, though. You might see one like the one below, with a number on it. This one is the cube root:

That's more advanced than what we're dealing with here, but know that if there's no number, then it's implied to be a square root.

Let's say we see this:

We'd call this the square root of 4. We might also say root 4 or radical 4. All of those phrases mean the same thing. We want to know what number squared is 4.

Of course, we know that 2 * 2 = 4. So our answer here is 2. You might think of it like a tree. A huge oak tree grows from a tiny acorn. Likewise, the root of 4 is 2.

Now, 2 * 2 isn't the only way to get 4. What if we did -2 * -2? That's also positive 4. But positive 2 is called the *principal root* and, certainly at this level of math, we only consider positive 2 to be our answer.

Let's practice! With small numbers, like 4, it's fairly easy to figure out the square root. But what about a bigger number? What if we had root 196? That's like a redwood tree. How can we figure out its square root?

We need to factor, or break the number into smaller parts. Because the factors will still be under our radical sign, it's smart to try to factor out perfect squares.

Let's start with 4. Does 4 go into 196? Yes! 196 / 4 is 49. So now we have root 4 * root 49. Well, hang on. 49 is familiar, isn't it? We saw that before when we were looking at perfect squares. 7 * 7 = 49. And we already know that root 4 is 2. So we can solve these easier, smaller radicals to get 2 * 7. 2 * 7 = 14. We should check to see if 14 * 14 is 196. Well, guess what? It is! We just solved root 196!

Let's try another. What about root 729? Ugh. That's bigger than the last one. If the last one was a redwood, what's this one? It's like a redwood on top of a redwood. That's a big tree. Anyway, let's factor.

Since factoring out a 1 doesn't get us anywhere, 4 is our smallest useful perfect square. That's why we start there. Does 4 go into 729? Not cleanly. The next smallest perfect square is 9. 9 does go into 729. 729 / 9 is 81. So now we have root 9 * root 81. Root 9 is 3. And hey, root 81 is familiar, isn't it? We know that the square root of 81 is 9. So we have 3 * 9, which is 27. Does 27 * 27 = 729? Yep! We did it!

To summarize, we started by reviewing squares. A square is a number multiplied by itself, or raised to the power of 2.

A square root is the inverse of squaring a number. We can say that the square root of *n*^2 is *n*.* n* * *n* gets us *n*^2.

A square root uses a radical sign. With small numbers, we can quickly figure out the square root. With larger numbers, we use factoring to break it up into smaller numbers until we get our answer.

After this video lesson, you should be able to:

- Explain what a square and a square root are
- Identify the sign used to indicate square root
- List examples of perfect squares
- Describe how to factor to find the square root of larger numbers

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Algebra I: High School20 chapters | 168 lessons | 1 flashcard set

- How to Find the Square Root of a Number 5:42
- Simplifying Square Roots When not a Perfect Square 4:45
- Simplifying Expressions Containing Square Roots 7:03
- Division and Reciprocals of Radical Expressions 5:53
- Radicands and Radical Expressions 4:29
- 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 High School Algebra: Radical Expressions

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