Student Academic Learning Services
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Converting between exponent and radical forms
Index (n)
Base (x)
𝑛
√𝑥
Exponent (1/n)
= 𝑥 1⁄𝑛
Radicand (x)
Things to Remember
If an index isn’t given then it is assumed to be 2.
The difference between the index and a number
being multiplied by the radical.
Example
2
√𝑥 means the same thing as √𝑥.
4√3 means 2 times the square root of 3.
4
Radical to Exponent
√3 means the 4th root of 3.
Steps
Examples
Take the index of the root and make it the
denominator of the exponent.
3
5
If there is an exponent somewhere, make it the
numerator.
3
� √4� = 43⁄5
Exponent to Radical
√x = x 1⁄3
�x 2 = x 2⁄3
Step
Examples
Take the denominator and make it the index of the
radical.
𝑥 1⁄3 = √𝑥
If the numerator is something other than 1, make it
the exponent of the radicand.
www.durhamcollege.ca/sals
√4 = 41⁄5
5
3
3
152⁄3 = �152
3
161⁄2 = √16
𝑦 5⁄2 = �𝑦 5
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905.721.2000 ext. 2491
This document last updated: 6/25/2012
Student Academic Learning Services
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Why convert?
Here are some suggestions about when and why you would want to convert between radical and
exponent form to solve a problem.
Advantages of radical form
Example
Easy to multiply and divide only if the indexes are
the same (simply put them in the same root).
�50 ∙ �2 = �100 = 10
3
3 135
3
=�
= √27 = 3
5
√5
√135
Easier to simplify a radical by factoring out perfect
powers.
3
√18 = √9√2 = 3√2
Would you have noticed you could do that if
it was written as 181⁄2 ?
Advantages of exponent form
Example
Easier to simplify by applying the exponent rules for
multiplying and dividing.
Simplify:
(That is, add the exponents together when
multiplying and subtract exponents when dividing
with the same base. The only difference is that the
exponents will be fractions.)
Easier to simplify when it is all raised to another
power.
(Remember that the rule for an exponent raised to
another exponent is to multiply the exponents
together e.g. (𝑥 2 )3 = 𝑥 6 ).
www.durhamcollege.ca/sals
4
4
�𝑥3 ∙ �𝑥5
𝑥2
√𝑥 3 ∙ √𝑥 5 𝑥 3⁄2 ∙ 𝑥 5⁄4
=
𝑥2
𝑥2
3⁄2+5⁄4−2⁄1
=𝑥
= 𝑥 6⁄4+5⁄4−8⁄4
= 𝑥 3⁄4
4
= �𝑥 3
4
2
Simplify: � �𝑥3 ∙ �𝑥5 �
4
2
2
� �𝑥3 � = �𝑥3⁄4 �
= 𝑥 (3⁄4)×2
= 𝑥 3⁄2
2
= �𝑥 3
Student Services Building (SSB), Room 204
905.721.2000 ext. 2491
This document last updated: 6/25/2012