Simplifying Radicals

Radicals

2 5 10

623

Simplifying Radicals

45 59

59

53

Express 45 as a product using a square number

Separate the product

Take the square root of the perfect square

Some Common Examples

12 3434

32

75 325325

35

18 2929

23

Harder Example

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

549

549

57

Addition and Subtraction

You can only add or subtract “like” radicals

535 54

773 72

236325 6322 You cannot add or subtract with

More Adding and Subtracting

83775

2437325

You must simplify all radicals before you can add or subtract

223735

22312

Multiplication

5327 1021

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

You multiply the outside numbers together and you multiply the numbers under the radical signs together

More Examples

756 425

6238 1816 Note that can be simplified

18

2916

2316

248

Try These

2463 1212 153107

15021

62521

6105

3412

324

Division

531012 53

1012

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

5

104

24

Division

35758 35

758

3

75

5

8

255

8

55

8

8

Important Points to Note

baab

bab

a

However baba

baba

Radicals can be separated when you have multiplication and division

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.

5

3

5

5

5

3

Here we are multiplying by 1

5

53

The denominator is now rational

More Rationalising Denominators

35

6

3

3

35

6

15

36 Simplify

Multiply by 1 in the form 3

3

5

32

Review Difference of Squares

))(( baba 22 bababa 22 ba

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

35

6

35

35

35

6

925

)35(6

Simplify

16

)35(6

Here we multiply by 5 – which is called the conjugate of 5 + 3

3

8

)35(3

Another Example

73

21

73

73

73

21

73

14763

Simplify

Here we multiply by the conjugate of which is

73 73

4

14763

4

14763

Try this one

352

56

352

352

352

56

9254

3518510302

Simplify

320

3523510302

The conjugate of

is 352 352

17

3523510302 See next

slide

Continuing

17

3523510302

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