Simplifying Radicals

2

Radicals

5 3  2 6 10

Simplifying Radicals

45   9  5 9   3 5 5 Express 45 as a product using a square number Separate the product Take the square root of the perfect square

Some Common Examples

12    2 4  3 4  3 3 75    5 25  3 25  3 3 18    3 9  2 9  2 2

Harder Example

245    7 49  5 49  5 5 Find a perfect square number that divides evenly into 245 by testing 4, 9, 16, 25, 49 (this works)

Addition and Subtraction

You can only add or subtract “like” radicals 3 5  3 5  4 5 7  7  2 7 5 2  3 6  3 2  2 2  3 You cannot add or subtract with 6

More Adding and Subtracting

75  7 3  8  25  3  7 3   5  12 3 3  7  2 3 2  2 2 You must simplify all radicals before you can add or subtract 4  2

Multiplication

Consider each radical as having two parts. The whole number out the front and the number under the radical sign.

7 2  3 5  21 10 You multiply the outside numbers together and you multiply the numbers under the radical signs together

More Examples

6  5 7  5 42 8 3  2 6  16  16 18 9  2  16  3  48 2 2 be simplified

Try These

3 6  4 2  12 12  12  24 4 3  3 7 10  3 15  21 150  21 25  6  105 6

Division

As with multiplication, we consider the two parts of the surd separately.

12 10  3 5  12 10 3 5  4 10 5  4 2

Division

8 75  5 3  8 5 75 3  8 5 75 3  8 5 25  8 5  5  8

Important Points to Note

ab



a



b a b



a



b

However

Radicals can be separated when you have multiplication and division

a



b



a



b a



b



a



b

Radicals

cannot

be separated when you have addition and subtraction

Rational Denominators

Radicals are irrational. A fraction with a radical in the denominator should to be changed so that the denominator is rational.

3 5  3 5  5 5 Here we are multiplying by 1  3 5 5 The denominator is now rational

More Rationalising Denominators

5 6 3  5 6 3  3 3  6 15 3  2 5 3 Multiply by 1 in 3 the form 3 Simplify

Review Difference of Squares

(

a



b

)(

a



b

)  

a

2 

ab



ab



b

2

a

2 

b

2 When a radical is squared, it is no longer a radical. It becomes rational. We use this and the process above to rationalise the denominators in the following examples.

More Examples

5  6 3  5  6 3  5  5  3 3  6 ( 5  25  9 3 )   6 ( 5  3 ) 3 ( 5 16  3 ) 8 Here we multiply by 5 – 3 which is called the conjugate of 5 + 3 Simplify

Another Example

1  3  2 7  1  3  2 7  3  3  7 7     3 3   3  6  3  7 7  6   4 7  6  7  14 14 14 4 Here we multiply by the conjugate is 3  7 Simplify

Try this one

2 6  5 5  3  2 6  5 5  3  2 2 5  5  3 3 The conjugate of 2 5  3 is 2 5  3  2 30  10 4 5  25  18  5 9 3 Simplify   2 2 30  10 5  20  3 3 30  10 5 17  3 2  5 2  5 3 3 See next slide

Continuing

 2 30  10 5  3 17 2  5 3

We wish to thank our supporters: