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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
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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
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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

12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Practice Key Terms 6

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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