Simplifying Square Root Expressions

Numbers with a RootRadical numbers are typically irrational numbers

(unless they simplify to a rational number). Our calculator gives:

But the decimal will go on forever and not repeat because it is an irrational number. For the exact answer just use:

1.41421

2

2

Some radicals can be simplified similar to simplifying a fraction.

Radical Product Property

ONLY when a≥0 and b≥0

For Example:

a b ab

1449 16 9 16

129 16 3 4 Equal

12

Perfect Squares

The square of whole numbers.

1 , 4 , 9 , 16 , 25 , 36 , 49 , 64 , 81 , 100 ,

121, 144 , 169 , 196 , 225, etc

Simplifying Square Roots1. Check if the square root is a whole number

2. Find the biggest perfect square (4, 9, 16, 25, 36, 49, 64) that divides the number in the root

3. Rewrite the number in the root as a product

4. Simplify by taking the square root of the perfect square and putting it outside the root

5. CHECK!

Note: A square root can not be simplified if there is no perfect square that divides it. Just leave it alone.

ex: √15 , √21, and √17

Simplifying Square RootsWrite the following as a radical (square root) in simplest

form:

6 236 272 36 2

3 39 327 9 3

5 4 2 5 16 25 32 5 16 2 20 2

36 is the biggest perfect square that divides 72.

Rewrite the square root as a product of roots.

Simplify.

Ignore the 5 multiplication until the end.

Simplifying Square RootsSimplify these radicals:

) 16 ) 8

) 7 ) 75

)4 63 ) 128

A B

C D

E F

4 2 2

5 3

12 7 8 2

Adding and Subtracting Radicals

Simplify the expressions:

2 3 2

. 2 3 2 4 3a 2 3 4 3 2

4 2 3 24 2 9 2

. 4 2 18b 4 2 9 2

7 2

Treat the square roots as variables, then combine like

terms ONLY.

Always simplify a radical first.

33 2

22

Multiplication and Radicals

Simplify the expression:

28 10 15

7 10 4 15

Conclusion: Multiply the numbers outside of the square root, then multiply the numbers inside of the square root. Then

simplify.

7 4 10 15 Use the Commutative Property

to Rewrite the expression.

Simplify and use the Radical Product Property Backwards.

28 150If possible, simplify more. 28 25 6

28 5 6140 6

Distribution and RadicalsRewrite the expression:

66 6 2

5 6 4 3 3 6 2 3

Remember: Multiply the numbers outside of the square root, then multiply the numbers

inside of the square root. Then simplify.

3√6 -2√3

5√6

4√3

15√36 90

-10√18 -30√2

12√18 36√2

-8√9 -24

Combine like terms.

90 30 2 36 2 24 Find the Sum.

Fractions and RadicalsSimplify the expressions:

4 122.b

4 4 32

There is nothing to simplify because the

square root is simplified and every term in the fraction

can not be divided by 10.

4 2 32

2 2 3

2

2 3

5 710.a 15 180

9.c

15 36 59

15 6 59

3 5 2 5

3 3

5 2 53

Make sure to simplify the

fraction.

Radical Quotient Property

ONLY when a≥0 and b≥0

For Example:

a a

b b

8464

1664

16

26416 4

Equal

2

The Square Root of a FractionWrite the following as a radical (square root) in simplest

form:

3

4

3

4

3

2

Take the square root of the numerator and the denominator

Simplify.

Rationalizing a DenominatorThe denominator of a fraction can not contain a radical. To

rationalize the denominator (rewriting a fraction so the bottom is a rational number) multiply by the same radical.

Simplify the following expressions:

5 2

2

2

5 2

25

2

2

2

6 3

5 3

2

6 3

5 36

5 3

3

3

6 3

15

3 2 3

3 5

2 3

5

WARNINGIn general:

a b a b For Example:

59 16 25

79 16 3 4

Not Equal