Operations with Radicals: Intermediate Algebra
Look At:
n
x is called the radicand
n is called the index
n
x is called the radical expression
is called the radical sign (or symbol)
x
Simplified Form of a Radical:
The purpose of having a simplified form for radical expressions is the same as that for
fractions. We all want to end up with our answers in the same form.
A radical expression is in simplified form when:
1. There are no factors in the radicand greater than or equal to the index. In other
words, the power of the factors in the radicand must be less than the index.
Simplified Radical
3
x
Not Simplified Radical
2
2
2. There are no radicals in the denominator.
Simplified Radical
x
5
3. There are no fractions in the radicand.
Simplified Radical
x
5
x3
Not Simplified Radical
x
5
Not Simplified Radical
x
5
To simplify radical expressions, we use the following relationships:
a na
n
n
=
ab = n a • n b
b nb
Before we actually simplify any radical expressions, let me try to convince you that the
previous relationships are true.
Look at:
Using the relationship n ab = n a • n b ,
100
2
= 2 100
100
= 2 (10) 2
= 2 4 • 25
=10
= 2 4 • 2 25
= 2•5
= 10
Notice that we get the same answer.
Look at:
Using the relationship
3
3
8
a
=
b
n
a
n
b
8
= 3 (2) 3
=3
216
27
3
=2
=
216
=
n
3
27
3
(6) 3
3
(3) 3
6
3
=2
Notice that we again get the same answer.
=
This seems like a waste of time in the previous problems because it is. The first problem
was a perfect square and the second problem was a perfect cube. There were no factors
to be left in the radicand. We use the above relationships when factors are to remain in
the radicand.
Even though I did not show it above, I mostly present simplification using a factoring
strategy. It may seem long and drawn out, but my students have the most success using
the following technique:
Simplify each radical expression. Issue: index is lower than powers of factors.
1. 3 125
=3 5•5•5
The index tells me to bring one factor outside of the radical for every group of three.
=5
2. 3 16
= 3 2 • 2 • 2 • 2 There is one group of three, with one factor remaining inside.
=2 3 2
3. 36
= 2 36
=2 2• 2•3•3
=2•3
=6
4.
48 x 4 y 7
4
=4 2• 2• 2• 2•3• x • x • x • x • y • y • y • y • y • y • y
=2• x • y • 4 3• y • y • y
= 2 xy
4
3y3
The issue of radical in the denominator is resolved by using a process called:
Rationalizing the Denominator.
In order to rationalize the denominator of any radical expression, multiply the
denominator and numerator by the radical expression necessary to create enough
factors so that the radical will disappear.
Simplify each radical expression. Issue: radical in the denominator.
5
1.
3
=3
4
5
3
2 in
2•2
the numerator and denominator. The reason I have to multiply it in the numerator goes
back to basic fraction rules. Whatever is multiplied in the denominator, must also be
multiplied in the numerator.
=3
=3
=
I notice that I need another factor of 2. This prompts me to multiply by
3
5
•3
2•2
5•3 2
2
2
2•2•2
5 32
2.
2
5
6x 2
2
=2
=2
=2
=2
5
2•3• x • x
2
5
I notice that I need one factor each of 2 and 3. The “ x -factor” is fine.
2
•2
2•3• x • x
2
5•2•3
2•3
2•3
2•3• x • x • 2•3
2
2•3•5
2• 2•3•3• x • x
2
30
2•3• x
30
=
6x
=
What about when you have a fraction inside of a radical? Split into two dividing
radical expressions, then rationalize the denominator, if necessary.
Simplify each radical expression. Issue: radicand is a fraction.
2
9
1.
=2
2
=2
2
9
2
9
2
=2
2
3•3
2
=
3
2
3
2.
=2
2
3
2
2
=2
2
=2
2
=2
3
2
2
•2
3
3
2•3
3•3
6
=
3
6
=
3
2
3
To Multiply Two Radical Expressions:
a n x • b n y = ab n xy
The radical factors must have the same index in order to combine.
Multiply and simplify.
1. 6 • 10
=2 2•3 •2 2•5
=2 2• 2•3•5
=2 2 3•5
= 2 15
2. 2 x
y• 3
3
3
x4
=2• x • 3 y •3• 3 x • x • x • x
=2• x •3• 3 y • x • x • x • x
=2• x •3• x • 3 y • x
= 6x 2
3.
=
=2
=2
=
3
xy
2( 3 − 2)
2• 3− 2• 2
2•3 − 2 2•2
6 −2
6 −2
To Add or Subtract Two Radical Expressions:
a n x + b n x = ( a + b) n x
a n x − b n x = ( a − b) n x
In order to combine two adding or subtracting radical expressions, the radical
factors must be identical – in radicand and index. To identify “like radicals,” each
radical expression must be in simplest form.
Add or subtract.
1. 4 xy − 7 xy
= − 3 xy
y +8
x3 y
2. 10 x
3
= 10 x
3
y +8
= 10 x
3
y +8• x•3 y
= 10 x
3
y + 8x
= 18 x
3
y
3
3
x•x•x• y
3
y