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Section 4.3
Terminology
Entire Radical: a radical in the form
Examples :
35 ,
x
68
Mixed Radical: a radical in the form a x
Examples : 3 5 ,  4 21
Part 1:
Entire Radicals to Mixed Radicals
72
 36  2
6 2
1. Find the LARGEST
perfect square that is a
factor of the radicand.
2. Rewrite the radicand
as a product of its
largest square and
another number.
3. Take the square root of
the perfect square and
write the answer as a
mixed radical
Remember the Perfect Squares:
1, 4 , 9, 16, 25, 36, 49, 64, 81, 100…
Examples
50
288
150
48
Examples
50  5 2
25  2
150  5 6
25  6
288  12 2
144  2
48  2 12
4  12
Make sure
your radical
is in lowest
terms!
48  2 12
4 12
4
3
2 2 3
4 3
HOW COULD WE HAVE
AVOIDED THIS PROBLEM?
48  4 3
16  3
We should have written the radicand as a
product of its largest square and another
number. Not just any square! Both methods
will get you full marks, as long as your final
answer has the radicand in its lowest terms!
6 27
4 20
5 80
Simplifying with indices greater
than 2
3
54
3
16
4
80
Part 2:
Mixed Radicals to Entire Radicals
3 6  9 6
 54
Try this one:
4 5  16  5
 80
1. Square the number
outside of the radical
and write it under a
radical sign.
2. Multiply the two
numbers together and
put the answer under a
radical sign.
More examples of mixed to entire
radicals 
6 27
3
2 2
5 2
Try this TOUGHIE! CONVERT
FROM MIXED TO ENTIRE
How can you rewrite 6 as a 4th root??
4
6 5
 1296  5
4
4
 6480
4
Just calculate 64 (6x6x6x6) and
place it under the 4th root sign!
REMEMBER!
If you are not confident that this is the answer,
always check by evaluating both numbers in
your calculator. They should match.