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Operations With Radicals

Addition, subtraction, multiplication and division with radicals can be accomplished by using the laws and rules for radicals. Add and subtract like radicands just like with a variable expression. To multiply and divide use the product, division and simplification laws. Division has an additional step (rationalization) to make sure that the final answer does not have a radical in the denominator.

Let's start by reviewing the simplification process using the following two laws.

Law 1 ab n = a n • b n     Law 2 a n n =a

1. 32 3 = 8 3 • 4 3 =2 4 3 Use ab n = a n • b n and simplify

2. 7 2 =7      Use a n n =a

3. x 4 4 =x      Use a n n =a

4. x 5 y 6 z 3

= x 3 3 . x 2 3 . y 3 3 . y 3 3 . z 3    Use ab n = a n • b n to break apart products         and strategically identify perfect cubes.

= x. x 2 3 .y.y. z 3 simplify using: a n n =a

= x•y•y• x 2 3 • z 3 regroup using commutative property

= x y 2 x 2 z 3      use Law 1 in reverse to put product back under radical


Addition and Subtraction - index and radicand MUST be the same

Use the rule  a  x n +b  x n =(a+b)  x n (Add the coefficients of "like" terms)


1. 3 x +4 x =(3+4) x =7 x

2. 4  xy 5 -5  xy 5 =(4-5) xy 5 =-  xy 5

3. -2 x +6  x 3 cannot be simplified indices are not the same.

4. 10 x -5 y   cannot be simplified radicand is not the same

5. 7  y 3 -8  y 3 +12  y 3 =(7-8+12)  y 3 =11  y 3

6. 2 + 50   has a different radicand BUT we can simplify 50

= 2 + 25 . 2 Use ab n = a n • b n

= 2 +5 2 Now there are "like" terms

= (1+5) 2    Use a  x n +b  x n =(a+b)  x n

= 6 2


Multiply - Index must be the same

ab n = a n • b n  Reversed a n • b n = ab n



1. 3 • 2 = 6

2. 4  5 3 •3 4 3

= 4•3• 5 3 • 4 3     regroup using commutative property of multiplication

= 12 20 3  use the product law reversed a n • b n = ab n


3. 2 • 10

= 2 • 2 • 5       rewrite 10  as  2 • 5 to take advantage of the law below

= 2 2 • 5 a n n =a

= 2 5


4. 3 8xy 3 .2 x 2 y 3 3

= 3.2 8 x 3 y 3 y 1 3 use the product law reversed a n • b n = ab n        to combine under one radical and strategically look for        cubed terms ( 8 = 23 and x . x2 = x3).

= 3.2.2.x.y. y 3   use rule a n n =a to simplify

= 12xy y 3  simplify


Division - Index must be the same

a b n = a n b n



1. 2 81 4

= 2 4 81 4 use   a b n = a n b n

= 2 4 3 simplify 81 4 to 3

2. 9 5

= 9 5 use a b n = a n b n

= 3 5 simplify 9 to 3


The denominator has a radical so it is not completely simplified.


3 5  . 5 5 Rationalize the denominator by multiplying the numerator and the    denominator by the denominator

= 3 5 5


In order to use the operations of addition, subtraction, multiplication and division with radicals, the rules/laws for radicals must be memorized and fluency with simplifying radicals is a must.



Related Links:
Math
algebra
Simplifying radical expressions
Simplifying radical expressions with variables
Adding radical expressions
Simplifying Radicals Worksheets
Radical Form to Exponential Form Worksheets


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