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Section 4-3: Mixed and Entire radicals
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Just as fractions have “lowest terms”, …
… radicals can have “simplest form”
Just as fractions can be mixed numbers or improper fractions …
… radicals can also be “mixed” or “entire”.
Eg.
𝟕
𝟒
=
𝟑
𝟏𝟒
or
𝟏𝟎
𝟗
𝟏
= 𝟏𝟗
ENTIRE radical
MIXED radical
√𝟐𝟒
𝟐√𝟔
Because:
√𝟐𝟒 = √𝟒 × 𝟔 = √𝟒 × √𝟔 = 𝟐 × √𝟔 = 𝟐√𝟔
We calculate √𝟒 because we know the
EXACT answer = 2
We leave √𝟔 alone because it is an
irrational number (√𝟔 = 𝟐. 𝟒𝟒𝟗𝟒𝟖𝟗 …
Just as you must ALWAYS write fractions in lowest terms … you
are also supposed to ALWAYS write radicals in simplest terms.
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The fact that we can SPLIT the √𝟐𝟒 into √𝟒 × √𝟔 is called the
“multiplication property of radicals”.
Works with 3√𝒓𝒐𝒐𝒕𝒔 (and all roots):
e.g. Simplify:
3
√𝟐𝟒

3
= √𝟖 × 𝟑
3
= 𝟐 × √𝟑
3
so write 𝟐 √𝟑
You have to try to find a ‘factor’ that has a √𝒓𝒐𝒐𝒕 .
3
You can use prime factorization instead if you get stuck.
e.g. Simplify:
3
√𝟐𝟒
3
= √𝟐 × 𝟐 × 𝟐 × 𝟑
Since the 2 repeats
three times, then
2 is a cube-root
‘answer’.
Write:
3
𝟐 √𝟑
Your calculator will NOT simplify radicals for you.
e.g. 1 page 215: Simplify
𝒂) √𝟖𝟎 ,
3
𝒃) √𝟏𝟒𝟒,
4
𝒄) √𝟏𝟔𝟐
2
a)
Since this is a SQUARE root …
… look for factors that appear
twice.
√𝟖𝟎
So simplified answer: 𝟐 × 𝟐 × √𝟓
𝟒√𝟓
Or you can write √𝟖𝟎 = √𝟐 × 𝟐 × 𝟐 × 𝟐 × 𝟓 = √𝟐 × 𝟐 × √𝟐 × 𝟐 × √𝟓
= 𝟐 × 𝟐 × √𝟓
= 𝟒√𝟓
b)
3
√𝟏𝟒𝟒
** Don’t be fooled into thinking this =12
Do a factor tree.
3
√𝟏𝟒𝟒
𝟑
3
So √𝟏𝟒𝟒 = √𝟐 × 𝟐 × 𝟐 × 𝟐 × 𝟑 × 𝟑
3
and look for factors in 3’s.
…since it’s a 3√𝒓𝒐𝒐𝒕.
Thus = 𝟐 × √𝟐 × 𝟑 × 𝟑 = which simplifies to
3
𝟐 √𝟏𝟖
3
c)
4
√𝟏𝟔𝟐
It’s a 4th root … so look for quadruples: 𝒔𝒊𝒎𝒑𝒍𝒊𝒇𝒊𝒆𝒅 = 𝟑 √𝟐
4
BE CAREFUL!!!
For example:
But NO!
But NO!
DO THE FACTOR TREE!!
√𝟐𝟎𝟎 looks like √𝟒 × 𝟓𝟎 so I write: 𝟐√𝟓𝟎
√𝟐𝟎𝟎 also looks like √𝟐𝟓 × 𝟖 so Jane writes: 𝟓√𝟖
√𝟐𝟎𝟎 also makes √𝟐 × 𝟏𝟎𝟎
and you write: 𝟏𝟎√𝟐
Who is correct?? The LOWEST AND SIMPLEST number in radical.
So YOU are right.
How to be RIGHT every time?
USE FACTOR TREE!
Example #2 page 216) There are MANY radicals that can’t be
simplified.
Such as:
√𝟐𝟔
and
√𝟑𝟎
4
No factor of these is a SQUARE.
and
√𝟐𝟔
√𝟑𝟎
Since both are SQUARE roots … you’re looking for doubles. There
are no doubles … so there is no simplification possible.
What about going the other direction?
Given a MIXED radical …
… write the ENTIRE radical
Example 3 page 217)
[a]
Convert the following to ‘entire’ radicals.
𝟒√ 𝟑
√𝟒 × 𝟒 × √𝟑
It is a SQUARE root sign [2]
so you write the 4 twice!!
If it was a cube root, you’d
write it 3-times. Etc.
= √𝟏𝟔 × √𝟑
√𝟒𝟖
3
[b] 𝟑 √𝟐
3
3
= √𝟑 × 𝟑 × 𝟑 × √𝟐
5
3
3
= √𝟐𝟕 × √𝟐
3
= √𝟓𝟒
𝟓
[c]
𝟐 √𝟐
If you ‘shortcut’ … just do it correctly!!
𝟓
= √𝟐 × 𝟐 × 𝟐 × 𝟐 × 𝟐 × 𝟐
𝟓
= √𝟔𝟒
ASSIGNMENT 4-3:
 CYU #1 − 3: pages 215−217
 Then page 218 & 219:
 #3, 4aceg, 5aceg, 6, 7, 9, 10ace, 11ace, 12ace, 13−16,
17a, 18a, 19, 20, 21, 22a, 23a.
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