# 1.3 Radicals and rational exponents  (Page 3/11)

 Page 3 / 11

## Adding square roots

Add $\text{\hspace{0.17em}}5\sqrt{12}+2\sqrt{3}.$

We can rewrite $\text{\hspace{0.17em}}5\sqrt{12}\text{\hspace{0.17em}}$ as $\text{\hspace{0.17em}}5\sqrt{4·3}.\text{\hspace{0.17em}}$ According the product rule, this becomes $\text{\hspace{0.17em}}5\sqrt{4}\sqrt{3}.\text{\hspace{0.17em}}$ The square root of $\text{\hspace{0.17em}}\sqrt{4}\text{\hspace{0.17em}}$ is 2, so the expression becomes $\text{\hspace{0.17em}}5\left(2\right)\sqrt{3},$ which is $\text{\hspace{0.17em}}10\sqrt{3}.\text{\hspace{0.17em}}$ Now we can the terms have the same radicand so we can add.

$10\sqrt{3}+2\sqrt{3}=12\sqrt{3}$

Add $\text{\hspace{0.17em}}\sqrt{5}+6\sqrt{20}.$

$13\sqrt{5}$

## Subtracting square roots

Subtract $\text{\hspace{0.17em}}20\sqrt{72{a}^{3}{b}^{4}c}\text{\hspace{0.17em}}-14\sqrt{8{a}^{3}{b}^{4}c}.$

Rewrite each term so they have equal radicands.

$\begin{array}{ccc}\hfill 20\sqrt{72{a}^{3}{b}^{4}c}& =& 20\sqrt{9}\sqrt{4}\sqrt{2}\sqrt{a}\sqrt{{a}^{2}}\sqrt{{\left({b}^{2}\right)}^{2}}\sqrt{c}\hfill \\ & =& 20\left(3\right)\left(2\right)|a|{b}^{2}\sqrt{2ac}\hfill \\ & =& 120|a|{b}^{2}\sqrt{2ac}\hfill \end{array}$
$\begin{array}{ccc}\hfill 14\sqrt{8{a}^{3}{b}^{4}c}& =& 14\sqrt{2}\sqrt{4}\sqrt{a}\sqrt{{a}^{2}}\sqrt{{\left({b}^{2}\right)}^{2}}\sqrt{c}\hfill \\ & =& 14\left(2\right)|a|{b}^{2}\sqrt{2ac}\hfill \\ & =& 28|a|{b}^{2}\sqrt{2ac}\hfill \end{array}$

Now the terms have the same radicand so we can subtract.

$120|a|{b}^{2}\sqrt{2ac}-28|a|{b}^{2}\sqrt{2ac}=92|a|{b}^{2}\sqrt{2ac}$

Subtract $\text{\hspace{0.17em}}3\sqrt{80x}\text{\hspace{0.17em}}-4\sqrt{45x}.$

$0$

## Rationalizing denominators

When an expression involving square root radicals is written in simplest form, it will not contain a radical in the denominator. We can remove radicals from the denominators of fractions using a process called rationalizing the denominator .

We know that multiplying by 1 does not change the value of an expression. We use this property of multiplication to change expressions that contain radicals in the denominator. To remove radicals from the denominators of fractions, multiply by the form of 1 that will eliminate the radical.

For a denominator containing a single term, multiply by the radical in the denominator over itself. In other words, if the denominator is $\text{\hspace{0.17em}}b\sqrt{c},$ multiply by $\text{\hspace{0.17em}}\frac{\sqrt{c}}{\sqrt{c}}.$

For a denominator containing the sum or difference of a rational and an irrational term, multiply the numerator and denominator by the conjugate of the denominator, which is found by changing the sign of the radical portion of the denominator. If the denominator is $\text{\hspace{0.17em}}a+b\sqrt{c},$ then the conjugate is $\text{\hspace{0.17em}}a-b\sqrt{c}.$

Given an expression with a single square root radical term in the denominator, rationalize the denominator.

1. Multiply the numerator and denominator by the radical in the denominator.
2. Simplify.

## Rationalizing a denominator containing a single term

Write $\text{\hspace{0.17em}}\frac{2\sqrt{3}}{3\sqrt{10}}\text{\hspace{0.17em}}$ in simplest form.

The radical in the denominator is $\text{\hspace{0.17em}}\sqrt{10}.\text{\hspace{0.17em}}$ So multiply the fraction by $\text{\hspace{0.17em}}\frac{\sqrt{10}}{\sqrt{10}}.\text{\hspace{0.17em}}$ Then simplify.

Write $\text{\hspace{0.17em}}\frac{12\sqrt{3}}{\sqrt{2}}\text{\hspace{0.17em}}$ in simplest form.

$6\sqrt{6}$

Given an expression with a radical term and a constant in the denominator, rationalize the denominator.

1. Find the conjugate of the denominator.
2. Multiply the numerator and denominator by the conjugate.
3. Use the distributive property.
4. Simplify.

## Rationalizing a denominator containing two terms

Write $\text{\hspace{0.17em}}\frac{4}{1+\sqrt{5}}\text{\hspace{0.17em}}$ in simplest form.

Begin by finding the conjugate of the denominator by writing the denominator and changing the sign. So the conjugate of $\text{\hspace{0.17em}}1+\sqrt{5}\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}1-\sqrt{5}.\text{\hspace{0.17em}}$ Then multiply the fraction by $\text{\hspace{0.17em}}\frac{1-\sqrt{5}}{1-\sqrt{5}}.$

Write $\text{\hspace{0.17em}}\frac{7}{2+\sqrt{3}}\text{\hspace{0.17em}}$ in simplest form.

$14-7\sqrt{3}$

## Using rational roots

Although square roots are the most common rational roots, we can also find cube roots, 4th roots, 5th roots, and more. Just as the square root function is the inverse of the squaring function, these roots are the inverse of their respective power functions. These functions can be useful when we need to determine the number that, when raised to a certain power, gives a certain number.

#### Questions & Answers

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
Ken Reply
proof
AUSTINE
sebd me some questions about anything ill solve for yall
Manifoldee Reply
how to solve x²=2x+8 factorization?
Kristof Reply
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
SO THE ANSWER IS X=-8
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
1KI POWER 1/3 PLEASE SOLUTIONS
Prashant Reply
hii
Amit
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Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
which of these functions is not uniformly continuous on 0,1
Basant Reply
solve this equation by completing the square 3x-4x-7=0
Jamiz Reply
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
Jean Reply
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
Rubben Reply
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
Pab Reply
More example of algebra and trigo
Stephen Reply
What is Indices
Yashim Reply
If one side only of a triangle is given is it possible to solve for the unkown two sides?
Felix Reply
cool
Rubben
kya
Khushnama
please I need help in maths
Dayo Reply
Okey tell me, what's your problem is?
Navin

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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