Surds - Mathematics Form 3 Notes

• Rational and Irrational Numbers
  • Rational Numbers
  • Irrational Numbers
• Surds
  • Order of surds
  • Simplification of Surds
  • Operation of Surds
  • Multiplication and Division of Surds
  • Rationalization of Surds
• Past KCSE Questions on the Topic.

Rational and Irrational Numbers

Rational Numbers

A rational number is a number which can be written in the form ^p/_q , where p and q are integers and q≠ 0.The integer’s p and q must not have common factors other than 1 .
Numbers such as 2, ¹/₂, ³/₄√4 are examples of rational numbers .Recurring numbers are also rational numbers.

Irrational Numbers

Numbers that cannot be written in the form ^p/_q .Numbers such as π √2√3 are irrational numbers. √2√3

Surds

Numbers which have got no exact square roots or cube root are called surds e.g. √2 , √8 √28 ,³√16 or ³√36

The product of a surd and a rational number is called a mixed surd. Examples are ;
2√3 , 4√7 and ¹/₃√2

Order of Surds

√3, √6 are surds of order two

³√2, ³√6 are surds of order three

⁴√2, ⁴√64 are surds of order four

Simplification of Surds

A surd can be reduced to its lowest term possible, as follows;

Example

Simplify

1. √18
2. √72

Solution

1. √18 = √9 x √2
   √9 × √2 = 3√2
2. 48 = √16 x √3
   √16 × √3 = 4√3

Operation of Surds

Surds can be added or subtracted only if they are like surds (that is, if they have the same value under the root sign).

Example 1

Simplify the following.

1. 3√2 + 5√2
2. 8√5 – 2√5

Solution

1. 3√2 + 5√2 = 8√2
2. 8 √5 – 2√5 = 6√5

Summary

√2 + √2 Let a = √2

Therefore √2 + √2 = a + a =2a

But a = √2
Hence √2 + √2 = 2√2

Multiplication and Division of Surds

Surds of the same order can be multiplied or divided irrespective of the number under the root sign.

Law 1: √a x √b = √ab When multiplying surds together, multiply their values together.

Example 1: √3 x √12 = √(3 x 12) = √36 = 6

Example 2: √7 x √5 = √35

This law can be used in reverse to simplify expressions…

Example 3: √12 = √2 x √6 or √4 x √3 = 2√3

Law 2: √a ÷ √b or ^√a/_√b = √(^a/_b) When dividing surds, divide their values (and vice versa).

Example 1: ^√12/_√3 = √(12 ÷ 3) = √4 = 2

Example 2: ^√6/_√8=^√3/_√4 = √(³/₄)

Law 3: √(a²) or (√a)² = a When squaring a square-root, (or vice versa), the symbols cancel each other out, leaving just the base.

Example 1: √12² = 12

Example 2: √7 x √7 = √7² = 7

Note:

• If you add the same surds together you just have that number of surds.
  For example √2 + √2 + √2= 3√2
• If a surd has a square number as a factor you can use law 1 and/or law 2 and work backwards to take that out and simplify the surd.
  For example √500 = √100 x √5 = 10√5

Rationalization of Surds

• Surds may also appear in fractions.
• Rationalizing the denominator of such a fraction means finding an equivalent fraction that does NOT have a surd on the bottom of the fraction (though itCAN have a surd on the top!).
• If the surd contains a square root by itself or a multiple of a square root, to get rid of it, you must multiply BOTH the top and bottom of the fraction by that square root value.
  Example 1:  6   x √7 = 6√7
                     √7  x √7      7
  
  Example 2: 6 + √2 x √3 = 6√3 + √2 x √3 = 6√3 + √6 = 6√3 + √6
                       2√3   x √3     2 x √3 x √3          2 x 3                 6
• If the surd on the bottom involves addition or subtraction with a square root, to get rid of the square root part you must use the ‘difference of two squares’ and multiply BOTH the top and bottom of the fraction by the bottom surd’s expression but with the inverse operation.
  Example 3:     7     x (2 –√2) = 14 – 7√2  = 14 – 7√2
                    2 + √2 x (2 –√2)     2² – (√2)²      4 – 2
  
  Notes on the ‘Difference of two squares’…
  
• Squaring(same operations)
  (2 + √2)(2 + √2)  = 2(2 + √2) + √2(2 + √2)
  = 4 + 2√2 + 2√2 + √2√2
  = 4 + 4√2 + 2 = 6 + √2 (still a surd)
• Multiplyin (opposite/inverse operations)
  (2 + √2)(2 - √2) = 2(2 – √2) + √2(2 – √2)
  = 4 – 2√2 + 2√2 – √2√2 (Note: – 2√2 and + 2√2 cancel out)
  = 4 – 2 = 2 (not a surd)
• In essence, as long as the operation in each brackets is the opposite, the middle terms will always cancel each other out and you will be left with the first term squared subtracting the second term squared.
  i.e. (5 + √7)(5 - √7) → 5² – (√7)² = 25 – 7 = 18

Example

Simplify by rationalizing the denominator

√2 + √3
√6 – √3

Solution

√2 + √3 × √6 + √3
√6 – √3 × √6 + √3

= √2( √6 + √3) + √3(√6 + √3) 
   √6(√6 + √3) – √3(√6 + √3)

= √12 + √6 + √18 + √9
   √36 + √18 – √18 – √9

= √4 x √3 + √6+ √9 x √2 + 3
                     6-3

=2√3 + √6 + 3√2 + 3
                3
Note

• If the product of the two surds gives a rational number then the product of the two surds gives conjugate surds.

Past KCSE Questions on the Topic.

1. Simplify (1 ÷ √3) (1 - √3)
   Hence evaluate     1    to 3 s.f. given that √3 = 1 .7321
                          1 + √3
2. If   √14     –      √14      = a√7 + b√2
     √7 – √2      √7 + √2
   Find the values of a and b where a and b are rational numbers.
3. Simplify as far as possible leaving your answer inform of a surd
          1           –           1     
   √14 - 2√3         √14 + 2√3
4. Given that tan 750 = 2 + √3, find without using tables tan 1 50 in the form p+q√m, where p, q and m are integers.
5. Without using mathematical tables, simplify
   √63 + √72
   √32 + √28
6. Simplify    3     +   1  leaving the answer in the form a + b √c, where a, b and c are rational
   numbers √5 – 2     √5