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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

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
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Stoney Reply
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Adin Reply
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biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
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Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
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Devang Reply
are you nano engineer ?
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
what is the actual application of fullerenes nowadays?
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
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