surds and indices

Contents

1. INTRODUCTION
2. SURDS
   1. PURE SURDS AND MIXED SURDS
   2. SIMILAR AND DISSIMILAR SURDS
   3. MULTIPLYING SURDS
   4. RATIONALIZATION OF SURDS
   5. COMPARISON OF SURDS
3. INDICES
   1. RULES OF INDICES
   2. COMPARISON OF INDICES

Surds and Indices

---

INTRODUCTION

We have come across numbers like , and 5⁴ in the concept of Number Systems. We read there that numbers such as … are irrational numbers. We also learnt about powers and exponents in Number Systems and Number Theory. In this chapter, we will focus on the rules that govern the mathematics of these numbers, and will learn how to manipulate them using the Laws of Indices.

SURDS

A surd is an ‘unresolved’ (meaning it still contains the symbol) mathematical expression of an n^th root. It can also be defined as an irrational number which is represented as the n^th root of a rational number.

For example, an irrational number which represents the n^th root of a positive rational number a is a surd and is represented as .

An alternate way of representing it is (this representation uses indices – we will discuss indices later in this chapter).

The symbol is called a radical, the base number a is called the radicand and n is the index of the radical, also known as the order of the surd.

Therefore are surds of order 2, 3 and 4 respectively.

is read as the third root (or the cube root) of 6 and is read as the fourth root of 7. If the index of the radical is not given, it is assumed to be 2.

For example, is read as the square root (or the second root) of 5.

IMPORTANT:

• Conditions for a number to be a surd are:

1. It is a positive irrational number

2. It is of the type, where a is a positive rational number

Hence, rational numbers like are not surds, because these are not unresolved. is equal to +2 and is equal to 3.

Again, is equal to or +20, hence it is not a surd.

Also, is not a surd, as the radicand itself is not a rational number.

PURE SURDS AND MIXED SURDS

In the surd , 5 is called the coefficient of the surd. When there is no coefficient in a surd, it is assumed that the coefficient is unity. Surds with unit coefficients are known as pure surds and surds with coefficients other than unity are known as mixed surds.

For example: are pure surds whereas are mixed surds.

The following rules of radicals are useful to simplify surds:

Example 1:

Convert the following into pure surds: Solution: We need to convert the given mixed surds into surds where the coefficient is 1. In other words, we have to ‘take the coefficient’ inside the root sign.

Example 2:

Convert the following into mixed surds: Solution: To convert a pure surd into a mixed surd, we have to see if there is any factor of the form an inside the nth root sign.

Example 3:

What is the square root of Solution: Now, Also, a + b + c = 3 + 45 + 25 = 73

and vice-versa

Explanation:

The same procedure can be followed to prove that

Example 4:

Solution: Multiplying the above two equations, we get, By comparing, we get, 100 = a3 + 3ab = a3 + 3a(a2 – 13) = a3 + 3a3 – 39a = 4a3 – 39a i.e. a(4a2 – 39) = 100 By trial and error, we get a = 4 and thus b = 42 – 13 = 3 Hence, the cube root of (100 – 51) is (4 –)

SIMILAR AND DISSIMILAR SURDS

Surds which have the same irrational part are known as similar surds (even if their coefficients are different).

Surds with different irrational parts are known as dissimilar surds.

For example, and are similar surds, whereas and are dissimilar surds.

We can add or subtract similar surds.

For example,

MULTIPLYING SURDS

Surds which have the same index can be multiplied without changing their index. The result of multiplication or division of surds having the same index will also have the same index.

For example, and can be multiplied, and the result will also have an index of 2. Similarly, and can be multiplied with the result having index 3. However, and cannot be multiplied directly as their indices are different; i.e. 2 and 3 respectively.

We multiply or divide such surds without changing the index. For example:

We can also multiply similar surds (since similar surds will have the same index). The result of multiplying/dividing two similar surds will always be an integer value.

Example 5:

Simplify the following: Solution:

RATIONALIZATION OF SURDS

The process of converting a surd to a rational number by multiplying it with a suitable number is called rationalization. To rationalize, we multiply the surd with a rationalizing factor. When the rationalizing factor is multiplied with the surd, we get a rational number.

For example, to rationalize we multiply it with (the rationalizing factor) to get and to rationalize we multiply it with to get

It is to be noted that there are always multiple rationalizing factors available. In the above example, we could have also multiplied with to rationalize it,

since

1. RATIONALIZING FACTOR OF A SUM OF SURDS

Consider the term which is the sum of two surds. The rationalizing factor for this is . When we multiply these two terms we get,

This brings us to the concept of the conjugate. The conjugate of is .

Similarly, the conjugate of is . The conjugate is the rationalizing factor (RF) for a sum of surds of the form .

2. USE OF RATIONALIZATION

Rationalization is mostly used to rationalize the denominator of an expression, if the denominator features a sum of surds.

Example 6:

Solution:

Rationalizing surds can often be simplified by converting the surds to their indices form.

Example 7:

Solution:

To rationalize a fraction such as the following steps are carried out:

1. Multiply the Numerator & Denominator by

2. Then multiply the Numerator & Denominator by [(x + y – z) – 2 ]

Explanation:

Thus, the expression is rationalized (since the denominator is a rational quantity).

Example 8:

Solution: Now,

COMPARISON OF SURDS

It is difficult to compare two surds of different indices. The strategy we follow is to change both surds to the same order. We can then compare them by the value of their radicands.

The new index of the two surds is the LCM of the original indices of the surds. The following example should make it clear.

Example 9:

Solution: The order of the surds is 2 and 3, hence we convert both the surds to surds of order 6 (which is the LCM of 2 and 3). Now we can say that Hence,

REMEMBER:

• The conversions of to and to is done using the rules of indices.

INDICES

Exponential notation is a convenient notation for repeated multiplication. It is generally used when a number is multiplied by itself several times.

Notation: If a is any rational number, then a a a ... n times = aⁿ. Here, a is called the base and n is called the exponent or index. The plural of the word index is indices.

For example,

4 4 = 4² is read as ‘the square of 4’ or ‘4 to the power 2’, and

5 5 5 = 5³ is read as ‘the cube of 5’ or ‘5 to the power 3’ and

6 6 6 6 = 6⁴ is read as ‘6 raised to the power 4’.

In general, aⁿ is read as ‘a raised to the power n’.

RULES OF INDICES

If a and b are non-zero rational numbers and m and n are rational numbers, then

Also, we can observe the following properties of indices.

We would now see a few applications of these rules.

Example 10:

Find the value of or simplify the following using the rules of indices: Solution:

Example 11:

Solve the following equations: Solution:

Example 12:

Calculate the values of x in the following equation: [JMET 2009] (1) 4, 9(2) 4, 9 (3) 4, 8(4) 7, 6 Solution: Since we have, x2 5x 36 = 0 (x 9)(x + 4) = 0 x = 9 or x = 4 Hence, option 1.

Example 13:

Given 2x = 8y + 1 and 9y = 3x – 9; the value of x + y is : [FMS 2010] (1) 18(2) 21 (3) 24(4) 27 Solution: 2x = 8y + 1 2x = (23) y + 1 2x = 23(y + 1) x = 3y + 3 …(i) Consider, 9y = 3x – 9 (32)y = 3 x – 9 32y = 3 x – 9 2y = x – 9 …(ii) Solving (i) and (ii) simultaneously, we get, x = 21 and y = 6 x + y = 21 + 6 = 27 Hence, option 4.

COMPARISON OF INDICES

To compare, we must try to equate the bases or the indices of the two numbers.

Example 14:

Which is greater? (i) 3245 or 6310 (ii) (22.5)27 or (7.5)54 Solution: (i) Now, 3245 = (34 22)5 = 320 210 Also, 6310 = (32 7)10 = 320 710 Since 710 > 210, we have 6310 > 3245 (ii) Here, let n = 27; then 2n = 54