# Roots for SQA National 5 Maths

1. Surds
2. Addition and subtraction of roots
3. Multiplication of surds
4. Division of surds
5. Simplifying surds
6. Rationalising denominators

Irrational numbers cannot be written as exact decimals or fractions. Surds are irrational numbers that can be represented as square/cube/… roots i.e. in surd form.

Only surds with the same number inside the square roots can be added together or subtracted from each other, unless it is possible to simplify the different roots to the same one. For example, 2√3 + 4√3 = 6√3.

To multiplying surds, find the product of the numbers outside the root and then the product of the numbers under the roots. If the same number is under the roots, the product of such roots is a whole number.

The division of surds should be completed by components − first divide the whole numbers, then divide the numbers under the roots.

In many calculations involving surds, the final number under the root can be simplified using the division and multiplication rules. The number under the root should be as small as possible.

To simplify a fraction with a surd in the denominator, multiply the top and bottom by the surd to rationalise the denominator.

# 1

Fully simplify √5(√125 − √5).

√5(√125 − √5) = √5(5√5 − √5) = √5 × 4√5 = 4 × 5 = 20

20

# 2

Simplify 9√3 × 4√3.

9√3 × 4√3 = 36√3² = 36 × 3 = 108

108

# 3

Simplify (8√6)/(2√4).

(8√6)/(2√4) = (4√6)/(√4) = (4√6)/2 = 2√6

2√6

# 4

√3 × √7 = √(3 × 7) = √21

√21

# 5

Simplify 6√10 + 3√2 − √10, if possible.

6√10 + 3√2 − √10 =
= (6 − 1)√10 + 3√2 =
= 5√10 + 3√2

5√10 + 3√2

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