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Roots for SQA National 5 Maths

Roots

This page covers the following topics:

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/โ€ฆ roots i.e. in surd form.

Surds

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, 2โˆš3 + 4โˆš3 = 6โˆš3.

Addition and subtraction of roots

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.

Multiplication of surds

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

Division of surds

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.

Simplifying surds

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

Rationalising denominators

1

Fully simplify โˆš5(โˆš125 โˆ’ โˆš5).

โˆš5(โˆš125 โˆ’ โˆš5) = โˆš5(5โˆš5 โˆ’ โˆš5) = โˆš5 ร— 4โˆš5 = 4 ร— 5 = 20

20

Fully simplify โˆš5(โˆš125 โˆ’ โˆš5).

2

Simplify 9โˆš3 ร— 4โˆš3.

9โˆš3 ร— 4โˆš3 = 36โˆš3ยฒ = 36 ร— 3 = 108

108

Simplify 9โˆš3 ร— 4โˆš3.

3

Simplify (8โˆš6)/(2โˆš4).

(8โˆš6)/(2โˆš4) = (4โˆš6)/(โˆš4) = (4โˆš6)/2 = 2โˆš6

2โˆš6

Simplify (8โˆš6)/(2โˆš4).

4

Simplify โˆš3 ร— โˆš7, leave your answer in surd form.

โˆš3 ร— โˆš7 = โˆš(3 ร— 7) = โˆš21

โˆš21

Simplify โˆš3 ร— โˆš7, leave your answer in surd form.

5

Simplify 6โˆš10 + 3โˆš2 โˆ’ โˆš10, if possible.

6โˆš10 + 3โˆš2 โˆ’ โˆš10 =
= (6 โˆ’ 1)โˆš10 + 3โˆš2 =
= 5โˆš10 + 3โˆš2

5โˆš10 + 3โˆš2

Simplify 6โˆš10 + 3โˆš2 โˆ’ โˆš10, if possible.

End of page

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