Back To Course

SAT Mathematics Level 2: Help and Review22 chapters | 225 lessons

Are you a student or a teacher?

Try Study.com, risk-free

As a member, you'll also get unlimited access to over 75,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Try it risk-freeWhat teachers are saying about Study.com

Already registered? Login here for access

Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*David Liano*

In this lesson, you will learn how to recognize square root notation. You will also learn properties of square roots and discover how to simplify expressions that include square roots.

Mathematics is the language of science, and like any language, there are terms and rules that you must master to speak it fluently. We'll explore the terms and rules that you'll need to follow any basic conversation involving **square root signs**. Like any sign, square root signs are trying to tell us something. They're telling us to find the square roots of whatever numbers are beneath them, and the numbers beneath a square root sign is called a **radicand**.

As for square roots, a number *r* is a square root of a number *x* if *r*^2 = *x.* Based on this definition, a positive number actually has two square roots: a positive number and the negative of that positive number. For example, the numbers 5 and -5 are square roots of 25 because 5^2 = 25 and (-5)^2 = 25. You can think of every positive root as having a negative, evil twin.

However, we're going to focus on the positive and limit our conversation to **principal square roots**, the positive square roots of positive numbers. For example, the principal square root of 25 is 5. Also, we won't cover the square roots of negative numbers, which would require a discussion of imaginary numbers, a conversation for another day. Now, let's learn to speak the language of square roots.

When speaking any language, it's good to know the simpler version of what you want to say. Perfect squares provide us with a way to find the simpler versions of square root expressions. A radicand is a **perfect square** if its principal square root is an integer. For example, 16 is a perfect square because its principal square root is 4. The number 26 is not a perfect square because its principal square root (about 5.1) is not an integer.

To visualize the idea of perfect squares, look at the following squares. You can display any perfect square (*r*^2) as an *r*-by-*r* grid, where *r* is an integer. For example, 4 equal-sized square units can be displayed as a square in a 2-by-2 grid. A 3-by-3 grid gives you 9 squares, a 4-by-4 square grid gives you 16 squares, and so on. However, you can't display a number that's not a perfect square, like 26, in this same fashion. There's no *r*-by-*r* square grid that'll give you 26 squares.

Recognizing perfect squares is important because it helps us simplify more complex square root expressions. Instead of saying, 'The square root of 16,' it's simpler to say '4.'

A square root expression is considered simplified once it meets two conditions:

1) The radicands have no perfect square factors other than 1

and

2) There are no square root signs in the denominator

What our first condition really means is we're looking for perfect squares under the square root sign. We can pull those perfect squares out from under the square root sign and write them as integers.

For example, to simplify the square root of 200, you'd want to find the perfect square factors of 200. We can write 200 as 2*100. Is 2 a perfect square? No, its principal square root is not an integer. Is 100 a perfect square? Yes, its principal square root is 10! Let's pull that 100 out from under the radical sign and write it as 10. So, the square root of 200 can be simplified to 10 times the square root of 2.

That one didn't seem too difficult, but what if we were asked to simplify the square root of 2 times the square root of the quantity 200 divided by 3. Don't be intimidated. In addition to recognizing perfect squares, there are two properties that'll help us accomplish this task.

If *a* is greater than 0 and *b* is greater than 0, then we can use the following properties to simplify square root expressions:

First, we have the **quotient property**, where the square root of *a* / *b* equals the square root of *a* divided by the square root of *b*.

We also have the **product property**, where the square root of *a***b* is the same as the square root of *a* times the square root of *b.*

We can use the quotient property to help with the square root of the quantity 200 divided by 3 part of our problem. Following this property, we can rewrite this part as the square root of 200 divided by the square root of 3. We know from earlier that the square root of 200 can be simplified to 10 times the square root of 2. So let's put that in our expression, too.

Now, we have the square root of 2 times 10 times the square root of 2 in the numerator and the square root of 3 in the denominator.

To simplify the numerator, we can use the product property to write the square root of 2 times the square root of 2 as the square root of the quantity 2 times 2. This gives us the square root of 4, so we now have 10 times the square root of 4 in the numerator. Don't forget that 4 is a perfect square, so we can rewrite the square root of 4 as 2. This gives us 20 in our numerator.

Are we done? No, remember that there can't be a square root sign in the denominator, so how do we get rid of that square root of 3?

We have to **rationalize the denominator**. This is a fancy way of saying that we need to multiply by some version of 1 to get rid of square root signs in the denominator. In this case, we multiply our entire expression by the square root of 3 divided by the square root of 3.

Multiplying our denominator by the square root of 3 gives us the square root of 3 times 3 (remember the product property). This gives us the square root of 9 in our denominator, which is just 3.

Our final, simplified version is 20 times the square root of 3 all over 3. The radicands have no perfect square factor other than 1 and there are no square root signs in the denominator, so we're done!

Are you feeling confident? Even if you don't know the solution yet, the answer should always be yes, so we're going to introduce one more concept: adding and subtracting square roots.

When adding and subtracting terms that have variables, you may remember that you're supposed to combine like terms. We apply this same concept to adding and subtracting square roots, except we combine terms that have the same radicand. Let's look at an example that puts this and everything else you've learned into practice. Feel free to pause the video at any time and work through the example yourself.

Simplify the following radical: 6 times the square root of the quantity 1 over 2 plus 4 times the square root of 18 minus 8 times the square root of 2.

That may look like a lot, but let's work on simplifying each term on its own. We can use the quotient property on our first term. This gives us 6 times the square root of 1 all over the square root of 2. The square root of 1 is just 1, so this is really 6 over the square root of 2.

To finish simplifying this term, we need to rationalize the denominator to get rid of the square root sign. Multiply this first term by the square root of 2 over the square root of 2, and you'll see that this term simplifies to 3 times the square root of 2.

Now, on to the second term. We can use the product property to help us find a perfect square. 4 times the square root of 18 becomes 4 times the square root of 2 times the square root of the perfect square 9. The square root of 9 is 3, so this gives us 4 times 3 times the square root of 2, or 12 times the square root of 2.

Okay, let's look at our final term, negative 8 times the square root of 2. Is there anything we can do to simplify it further? No, the radicand has no perfect squares, and there's certainly no square root sign in the denominator.

Our final step is to add and subtract these square roots. Remember that we can only combine terms that have the same radicand, but all of our terms have the radicand 2. This means we can combine all of them and rewrite our expression as the square root of 2 times the quantity 3 + 12 - 8. This just gives us 7 times the square root of 2, and we're finished!

As long as you take it step by step and remember the simplification rules, you'll soon speak the language of square roots fluently.

In the language of mathematics, **square root signs** tell us to find the square roots of whatever numbers are beneath them. The number beneath a square root sign is the **radicand**. Remember, a number *r* is a square root of a number *x* if *r*^2 = *x*.

Knowing how to speak about square root expressions means knowing how to put them in simple terms. You've expressed your square root expression in simplest terms once it meets two conditions:

1) The radicands have no perfect square factors other than 1

and

2) There are no square root signs in the denominator

In addition to recognizing perfect squares, there are two properties that help with simplifying radical expressions.

If *a* is greater than 0 and *b* is greater than 0, then we can use the **quotient property**, where the square root of *a* / *b* equals the square root of *a* divided by the square root of *b.*

We can also use the **product property**, where the square root of *a***b* is the same as the square root of *a* times the square root of *b.*

Also, don't forget to **rationalize the denominator**, or multiply by some version of 1 to get rid of the square root signs in the denominator.

Square root sign | tells us to find the square roots of whatever numbers are beneath them |

Radicand | the number beneath a square root sign |

Quotient property | the square root of a / b equals the square root of a divided by the square root of b |

Product property | the square root of a*b is the same as the square root of a times the square root of b |

Rationalize the denominator | multiply by some version of 1 to get rid of square root signs in the denominator |

Focus on concepts related to square roots during this lesson so that you can subsequently:

- Recognize a square root sign, a radicand and a perfect square
- Simplify and solve for a square root
- Use the quotient and product properties
- Add and subtract square roots

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackWhat teachers are saying about Study.com

Already registered? Login here for access

Did you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
7 in chapter 2 of the course:

Back To Course

SAT Mathematics Level 2: Help and Review22 chapters | 225 lessons

- Evaluating Square Roots of Perfect Squares 5:12
- Estimating Square Roots 5:10
- Simplifying Square Roots When not a Perfect Square 4:45
- Simplifying Expressions Containing Square Roots 7:03
- Discriminant: Definition & Explanation
- Principal Square Root: Definition & Example 5:02
- Square Root: Sign, Rules & Problems 10:15
- Adding Square Roots
- Subtracting Square Roots
- Go to Square Roots: Help and Review

- AFOQT Information Guide
- ACT Information Guide
- Computer Science 335: Mobile Forensics
- Electricity, Physics & Engineering Lesson Plans
- Teaching Economics Lesson Plans
- FTCE Middle Grades Math: Connecting Math Concepts
- Social Justice Goals in Social Work
- Developmental Abnormalities
- Overview of Human Growth & Development
- ACT Informational Resources
- AFOQT Prep Product Comparison
- ACT Prep Product Comparison
- CGAP Prep Product Comparison
- CPCE Prep Product Comparison
- CCXP Prep Product Comparison
- CNE Prep Product Comparison
- IAAP CAP Prep Product Comparison

- What's the Difference Between Polytheism and Monotheism?
- Ethnic Groups in America
- What Are the 5 Ws in Writing? - Uses & Examples
- Phenol: Preparation & Reactions
- Plant Life Cycle Project Ideas
- Medieval Castle Project Ideas
- Samurai Project Ideas
- Quiz & Worksheet - Solvay Process
- Quiz & Worksheet - Kinds of Color Wheels
- Quiz & Worksheet - Understanding Abbreviations
- Quiz & Worksheet - Act & Rule Utilitarianism Comparison
- Analytical & Non-Euclidean Geometry Flashcards
- Flashcards - Measurement & Experimental Design
- What is Differentiated Instruction? | Differentiated Instruction Resources
- Common Core ELA Standards | A Guide to Common Core ELA

- AP Physics 1: Exam Prep
- Explorations in Core Math - Geometry: Online Textbook Help
- NY Regents Exam - Chemistry: Help and Review
- Introduction to Political Science: Certificate Program
- DSST Computing and Information Technology: Study Guide & Test Prep
- Constructing Figures in Geometry
- Calculating & Simplifying Exponential Expressions
- Quiz & Worksheet - Types of Corporate Structures
- Quiz & Worksheet - Aggregate Supply Curve
- Quiz & Worksheet - Lead Qualification Process
- Quiz & Worksheet - Characteristics of Invertebrates
- Quiz & Worksheet - LM Curve in Macroeconomics

- What Is Organizational Structure of Management? - Types & Examples
- Chordata Digestive System
- New York State Physical Education Standards
- Scarcity Lesson Plan
- Persuasive Writing Activities
- Response to Intervention (RTI) in Florida
- WIDA Can Do Descriptors for Grade 1
- Activities for Kids with Cerebral Palsy
- 504 Plans in Georgia
- Reading Games for Kids
- Study.com's Teacher Edition
- ADHD Advocacy Groups

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject