# 1.3 Radicals and rational expressions

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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.

$\begin{array}{ccc}\hfill {a}^{2}+{b}^{2}& =& {c}^{2}\hfill \\ \hfill {5}^{2}+{12}^{2}& =& {c}^{2}\hfill \\ \hfill 169& =& {c}^{2}\hfill \end{array}$

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 $\text{\hspace{0.17em}}{4}^{2}=16,$ the square root of $\text{\hspace{0.17em}}16\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}4.\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is a positive real number, then the square root of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is a number that, when multiplied by itself, gives $\text{\hspace{0.17em}}a.\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}a.\text{\hspace{0.17em}}$ The square root obtained using a calculator is the principal square root.

The principal square root of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is written as $\text{\hspace{0.17em}}\sqrt{a}.\text{\hspace{0.17em}}$ The symbol is called a radical    , the term under the symbol is called the radicand    , and the entire expression is called a radical expression    .

## Principal square root

The principal square root    of $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is the nonnegative number that, when multiplied by itself, equals $\text{\hspace{0.17em}}a.\text{\hspace{0.17em}}$ It is written as a radical expression     , with a symbol called a radical    over the term called the radicand    : $\text{\hspace{0.17em}}\sqrt{a}.$

Does $\text{\hspace{0.17em}}\sqrt{25}=±5?$

No. Although both $\text{\hspace{0.17em}}{5}^{2}\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{\left(-5\right)}^{2}\text{\hspace{0.17em}}$ are $\text{\hspace{0.17em}}25,$ the radical symbol implies only a nonnegative root, the principal square root. The principal square root of 25 is $\text{\hspace{0.17em}}\sqrt{25}=5.$

## Evaluating square roots

Evaluate each expression.

1. $\sqrt{100}$
2. $\sqrt{\sqrt{16}}$
3. $\sqrt{25+144}$
4. $\sqrt{49}-\sqrt{81}$
1. $\sqrt{100}=10\text{\hspace{0.17em}}$ because $\text{\hspace{0.17em}}{10}^{2}=100$
2. $\sqrt{\sqrt{16}}=\sqrt{4}=2\text{\hspace{0.17em}}$ because $\text{\hspace{0.17em}}{4}^{2}=16\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{2}^{2}=4$
3. $\sqrt{25+144}=\sqrt{169}=13\text{\hspace{0.17em}}$ because $\text{\hspace{0.17em}}{13}^{2}=169$
4. $\sqrt{49}-\sqrt{81}=7-9=-2\text{\hspace{0.17em}}$ because $\text{\hspace{0.17em}}{7}^{2}=49\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}{9}^{2}=81$

For $\text{\hspace{0.17em}}\sqrt{25+144},$ can we find the square roots before adding?

No. $\text{\hspace{0.17em}}\sqrt{25}+\sqrt{144}=5+12=17.\text{\hspace{0.17em}}$ This is not equivalent to $\text{\hspace{0.17em}}\sqrt{25+144}=13.\text{\hspace{0.17em}}$ The order of operations requires us to add the terms in the radicand before finding the square root.

Evaluate each expression.

1. $\sqrt{225}$
2. $\sqrt{\sqrt{81}}$
3. $\sqrt{25-9}$
4. $\sqrt{36}+\sqrt{121}$
1. $15$
2. $3$
3. $4$
4. $17$

## 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 $\text{\hspace{0.17em}}\sqrt{15}\text{\hspace{0.17em}}$ as $\text{\hspace{0.17em}}\sqrt{3}\cdot \sqrt{5}.\text{\hspace{0.17em}}$ We can also use the product rule to express the product of multiple radical expressions as a single radical expression.

what is math number
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
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
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
hi vedant can u help me with some assignments
Solomon
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
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
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
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
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar