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By the end of this section, you will be able to:
  • Simplify expressions with square roots
  • Estimate square roots
  • Approximate square roots
  • Simplify variable expressions with square roots
  • Use square roots in applications

Before you get started, take this readiness quiz.

  1. Simplify: ( −9 ) 2 .
    If you missed this problem, review Multiply and Divide Integers .
  2. Round 3.846 to the nearest hundredth.
    If you missed this problem, review Decimals .
  3. Evaluate 12 d for d = 80 .
    If you missed this problem, review Evaluate, Simplify and Translate Expressions .

Simplify expressions with square roots

To start this section, we need to review some important vocabulary and notation.

Remember that when a number n is multiplied by itself, we can write this as n 2 , which we read aloud as n squared.” For example, 8 2 is read as “8 squared.”

We call 64 the square of 8 because 8 2 = 64 . Similarly, 121 is the square of 11 , because 11 2 = 121 .

Square of a number

If n 2 = m , then m is the square of n .

Modeling squares

Do you know why we use the word square ? If we construct a square with three tiles on each side, the total number of tiles would be nine.

A square is shown with 3 tiles on each side. There are a total of 9 tiles in the square.

This is why we say that the square of three is nine.

3 2 = 9

The number 9 is called a perfect square because it is the square of a whole number.

Doing the Manipulative Mathematics activity Square Numbers will help you develop a better understanding of perfect square numbers

The chart shows the squares of the counting numbers 1 through 15 . You can refer to it to help you identify the perfect squares.

A table with two columns is shown. The first column is labeled “Number” and has the values: n, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15. The second column is labeled “Square” and has the values: n squared, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.

Perfect squares

A perfect square is the square of a whole number.

What happens when you square a negative number?

( −8 ) 2 = ( −8 ) ( −8 ) = 64

When we multiply two negative numbers, the product is always positive. So, the square of a negative number is always positive.

The chart shows the squares of the negative integers from −1 to −15 .

A table is shown with 2 columns. The first column is labeled “Number” and contains the values: n, negative 1, negative 2, negative 3, negative 4, negative 5, negative 6, negative 7, negative 8, negative 9, negative 10, negative 11, negative 12, negative 13, negative 14, and negative 15. The next column is labeled “Square” and contains the values: n squared, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, and 225.

Did you notice that these squares are the same as the squares of the positive numbers?

Square roots

Sometimes we will need to look at the relationship between numbers and their squares in reverse. Because 10 2 = 100 , we say 100 is the square of 10 . We can also say that 10 is a square root of 100 .

Square root of a number

A number whose square is m is called a square root of m .

If n 2 = m , then n is a square root of m .

Notice ( −10 ) 2 = 100 also, so −10 is also a square root of 100 . Therefore, both 10 and −10 are square roots of 100 .

So, every positive number has two square roots: one positive and one negative.

What if we only want the positive square root of a positive number? The radical sign, 0 , stands for the positive square root. The positive square root is also called the principal square root .

Square root notation

m is read as “the square root of m .”

If m = n 2 , then m = n for n 0 .

A picture of an m inside a square root sign is shown. The sign is labeled as a radical sign and the m is labeled as the radicand.

We can also use the radical sign for the square root of zero. Because 0 2 = 0 , 0 = 0 . Notice that zero has only one square root.

The chart shows the square roots of the first 15 perfect square numbers.

A table is shown with 2 columns. The first column contains the values: square root of 1, square root of 4, square root of 9, square root of 16, square root of 25, square root of 36, square root of 49, square root of 64, square root of 81, square root of 100, square root of 121, square root of 144, square root of 169, square root of 196, and square root of 225. The second column contains the values: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, and 15.

Simplify: 25 121 .


(a) 25 Since 5 2 = 25 5 (b) 121 Since 11 2 = 121 11

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Simplify: 36 169 .

  1. 6
  2. 13

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Simplify: 16 196 .

  1. 4
  2. 14

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Every positive number has two square root s and the radical sign indicates the positive one. We write 100 = 10 . If we want to find the negative square root of a number, we place a negative in front of the radical sign. For example, 100 = −10 .

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
yes that's correct
I think
Nasa has use it in the 60's, copper as water purification in the moon travel.
nanocopper obvius
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
how nano science is used for hydrophobicity
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
analytical skills graphene is prepared to kill any type viruses .
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and $10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply

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