Contents Boardworks Ltd 2005* of 38

Introducing of surdsManipulating surdsSimplifying surdsAdding and subtracting surdsExpanding brackets containing surdsRationalizing the denominatorThe index lawsZero and negative indicesFractional indices

Numbers written in this form are called surds.When the square root of a number, for example 2, 3 or 5,is irrational, it is often preferable to write it with the root sign.= 1.3

When working with surds it is important to remember the following two rules:

Start by finding the largest square number that divides into 50.This is 25. We can use this to write:

Surds can be added or subtracted if the number under the square root sign is the same. For example:

When a fraction contains a surd as the denominator we usually rewrite it so that the denominator is a rational number.This is called rationalizing the denominator. For example:2

Simplify the following fractions by rationalizing their denominators.3528

More difficult examples may include surds in both the numerator and the denominator. For example:Simplify= 6+ 1

Simplify:a a a a a= a5a to the power of 5

a5 has been written using index notation.anThe number a is called the base.The number n is called the index, power or exponent.In general:

For example:a4 a2 =(a a a a) (a a)= a a a a a a= a6When we multiply two terms with the same base the indices are added.= a (4 + 2)In general:am an = a(m + n)

For example:a5 a2 =a34p6 2p4 =2p2= a (5 2)= 2p(6 4)When we divide two terms with the same base the indices are subtracted.In general:am an = a(m n)2

For example:(y3)2 =(pq2)4 =y3 y3 = (y y y) (y y y) = y6 pq2 pq2 pq2 pq2= p4 q (2 + 2 + 2 + 2)= p4 q8= p4q8When a term is raised to a power and the result raised to another power, the powers are multiplied.In general:= y32= p14q24(am)n = amn

Look at the following division:y4 y4 =1But using the rule that xm xn = x(m n)y4 y4 = y(4 4) =y0That means thaty0 = 1In general:a0 = 1(for all a 0)Any number or term divided by itself is equal to 1.

Write the following using fraction notation:This is the reciprocal of u.

Indices can also be fractional. For example:So= a1Using the multiplication rule:= a

THANK YOU.

**Explain that when the root of a number is irrational it is neater and more accurate to write it in surd form.You may wish to point out that surds can also be written as cube roots or any other root to the nth degree.**Explain that when a surd is written in its simplest form, the number under the square root sign must not contain any factors that are square numbers.*Compare this to adding like terms in algebra.**Ask students to suggest what the numerator and denominator of each fraction should be multiplied by before revealing the solutions.*Some students could benefit from seeing the individual steps involved in multiplying out the brackets in the numerator. This is shown in the working. Students could use a calculator to check that the two forms are equivalent.*Revise the use of index notation to mean n lots of a multiplied together. Explain that, although this definition does not make sense when n is zero, negative or fractional, we are able to use the laws of indices shown on the next three slides to extend index notation to all rational exponents.*Explain that the indices can only be added when the base is the same.*Explain that the indices can only be subtracted when the base is the same.**Explain that we can use the three basic rules of indices to define zero, negative and fractional indices, even though these dont make sense within the context of the original definition. For example, we couldnt define 34 as minus four threes multiplied together.Explain that the rule for the zero index is only true for non-zero values of a. 00 is undefined.**Students may remember from work done at GCSE that a to the power of is equivalent to a.Discuss the fact that when we square the square root of a number we end up with the original number.It may be worth noting that a is defined to be the positive rather than the negative square root of a.