Surds

Generally, for GCSE and A-level students, a surd is a number which contains part square root/cube root and part rational number, e.g., root2 = 1 * root2 and root3 = 1 * root3, where 1 is the rational number and root2 and root3 respectively are the square roots.

NOTE: Not every root (square or cube) is a surd, e.g., root2 and root3 are surds, but root4 is NOT a surd since this simplifies to 2.

Manipulating Surds: Expressing Square Roots in terms of Simpler Square Roots

Question 1) Express the following surds in their simplest form.

1a) root44

1b) root63

1c) root76

1d) root300

Answer 1)

1a) root44 = root11 * root4 = 2 root11

1b) root63 = root7 * root9 = 3 root7

1c) root76 = root19 * root4 = 2 root19

1d) root300 = root3 * root100 = 10 root3

Manipulating Surds: Multiplying Surds

This practice is rather straightforward ~ simply multiply the surds like you multiply numbers, e.g., root5 * root7 = root35.

NOTE: If you multiply two surds which contain the same number/expression inside the surd, this returns the number/expression, e.g., root3 * root3 = 3 and root11 * root11 = 11

Question 2) Simplify fully.

2a) root8 * root9

2b) 12 root8 * 6 root3

2c) (root6)⁴

Answer 2)

2a) root8 * root9 = 3 root8 = 3 * root4 * root2 = 6 root2

2b) 12 root8 = 12 * root4 * root2 = 24 root2

So, 12 root8 * 6 root3 = 24 root2 * 6root3 = 144 root6

2c) (root6)⁴ = root6 * root6 * root6 * root6 = 6 * 6 = 36

Manipulating Surds: Adding Subtracting Surds

This practice is rather like adding and subtracting like terms in algebraic simplification. Therefore, you can collect terms involving rational numbers and can collect terms involving roots of the same number.

NOTE: You cannot collect terms involving roots of different numbers, e.g., root3 and root5.

Question 3) Evaluate the following.

3a) root90 - root80

3b) 3 root12 - root75

3c) root125 + root20

Answer 3)

3a) root90 = root9 * root10 = 3 root10

root80 = root8 * root10 = (root4 * root2) * (root2 * root5) = 2 * 2 * root5 = 4 root5

So, root90 - root80 = 3 root10 4 root5

3b) 3 root12 = 3 * root3 * root4 = 3 * 2 * root3 = 6 root3

root75 = root25 * root3 = 5 root3

So, 3 root12 - root75 = 6 root3 5 root3 = root3

3c) root125 = root25 * root5 = 5 root5

root20 = root4 * root5 = 2 root5

So, root125 + root20 = 5 root5 + 2 root5 = 7 root5

Manipulating Surds: Rationalising the Denominator

COMMON QUESTION (GCSE)

The question of rationalising the denominator arises if we observe surds in the denominator of a fraction.

The idea is that if the denominator of a fraction takes the form x + y, where either or both x and y are surds, then we can rationalise this term by multiplying both the numerator and denominator by the fraction x y. The denominator thus becomes (x + y) (x y), which is rational.

NOTE: Multiplying both the numerator and denominator by the same amount implies that the value of the fraction is unchanged.

Question 4) Rationalise the following.

4a) 14 / root5

4b) 28 / root6

4c) 4 / (root3 1)

4d) (root2 + 7) / (root2 1)

Answer 4)

4a) 14 / root5 = (14 / root5) * (root5 / root5) = 14 root5 / 5

4b) 28 / root6 = (28 / root6) * (root6 / root6) = 28 root6 / 6 = 14 root6 / 3

4c) 4 / (root3 1) = (4 / (root3 1)) * ((root3 + 1) / (root3 + 1)) = (4 root3 + 4) / (3 + root3 - root3 1) = (4 root3 + 4) / 2 = 2 root3 + 2

4d) (root2 + 7) / (root2 1) = ((root2 + 7) / (root2 1)) * ((root2 + 1) / (root2 + 1)) = (2 + root2 + 7 root2 + 7) / (2 + root2 - root2 1) = 9 + 8root2

CRUCIAL POINTS

Simplify surds wherever possible to make them easier to work with

When rationalising the denominator, ensure to multiply BOTH the numerator and denominator by the same expression