<< Chapter < Page Chapter >> Page >
In this section students will:
  • Evaluate square roots.
  • Use the product rule to simplify square roots.
  • Use the quotient rule to simplify square roots.
  • Add and subtract square roots.
  • Rationalize denominators.
  • Use rational roots.

A hardware store sells 16-ft ladders and 24-ft ladders. A window is located 12 feet above the ground. A ladder needs to be purchased that will reach the window from a point on the ground 5 feet from the building. To find out the length of ladder needed, we can draw a right triangle as shown in [link] , and use the Pythagorean Theorem.

A right triangle with a base of 5 feet, a height of 12 feet, and a hypotenuse labeled c
a 2 + b 2 = c 2 5 2 + 12 2 = c 2 169 = c 2

Now, we need to find out the length that, when squared, is 169, to determine which ladder to choose. In other words, we need to find a square root. In this section, we will investigate methods of finding solutions to problems such as this one.

Evaluating square roots

When the square root of a number is squared, the result is the original number. Since 4 2 = 16 , the square root of 16 is 4. The square root function is the inverse of the squaring function just as subtraction is the inverse of addition. To undo squaring, we take the square root.

In general terms, if a is a positive real number, then the square root of a is a number that, when multiplied by itself, gives a . The square root could be positive or negative because multiplying two negative numbers gives a positive number. The principal square root    is the nonnegative number that when multiplied by itself equals a . The square root obtained using a calculator is the principal square root.

The principal square root of a is written as a . The symbol is called a radical    , the term under the symbol is called the radicand    , and the entire expression is called a radical expression    .

The expression: square root of twenty-five is enclosed in a circle. The circle has an arrow pointing to it labeled: Radical expression. The square root symbol has an arrow pointing to it labeled: Radical. The number twenty-five has an arrow pointing to it labeled: Radicand.

Principal square root

The principal square root    of a is the nonnegative number that, when multiplied by itself, equals a . It is written as a radical expression     , with a symbol called a radical    over the term called the radicand    : a .

Does 25 = ± 5 ?

No. Although both 5 2 and ( −5 ) 2 are 25 , the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is 25 = 5.

Evaluating square roots

Evaluate each expression.

  1. 100
  2. 16
  3. 25 + 144
  4. 49 81
  1. 100 = 10 because 10 2 = 100
  2. 16 = 4 = 2 because 4 2 = 16 and 2 2 = 4
  3. 25 + 144 = 169 = 13 because 13 2 = 169
  4. 49 81 = 7 9 = −2 because 7 2 = 49 and 9 2 = 81
Got questions? Get instant answers now!
Got questions? Get instant answers now!

For 25 + 144 , can we find the square roots before adding?

No. 25 + 144 = 5 + 12 = 17. This is not equivalent to 25 + 144 = 13. The order of operations requires us to add the terms in the radicand before finding the square root.

Evaluate each expression.

  1. 225
  2. 81
  3. 25 9
  4. 36 + 121
  1. 15
  2. 3
  3. 4
  4. 17
Got questions? Get instant answers now!

Using the product rule to simplify square roots

To simplify a square root, we rewrite it such that there are no perfect squares in the radicand. There are several properties of square roots that allow us to simplify complicated radical expressions. The first rule we will look at is the product rule for simplifying square roots, which allows us to separate the square root of a product of two numbers into the product of two separate rational expressions. For instance, we can rewrite 15 as 3 5 . We can also use the product rule to express the product of multiple radical expressions as a single radical expression.

Questions & Answers

what are you up to?
Mark Reply
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
I dont know. lol
Jeffrey
please where is the equation
Miranda
ask nelson. lol
Jeffrey
answer and questions in exercise 11.2 sums
Yp Reply
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
Swadesh
what is a algebra
Jallah Reply
(x+x)3=?
Narad
6x
Obed
what is the identity of 1-cos²5x equal to?
liyemaikhaya Reply
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
Karl Reply
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
Aashish Reply
sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
Practice Key Terms 6

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask