Chapter 10 : Indices and Surds

Indices

If any number ‘a’ is multiplied 5 times, we say it as ‘ a raise to the power 5’ or a5.
Here ‘a’ is expressed as an exponent, where ‘a’ is the base and
5 is called the power or index of ‘a’.

So, we can write ‘a’ as any exponent ‘n’ which will be written as ‘an‘.
This can be written as ‘a to the power n’.

Example: 45 = 4 × 4 × 4 × 4 × 4 = 1024

Rules of Indices

1. _am × _an = _am+n

2. _am / _an = _am-n

3. (_am)n = _amn

4. a-m = 1 / am

5. _(ab)m = _am _bm

6. _a0 = 1

7. _a1 = a

Note:
1. If given that (_am) = (_an)

Case 1: if a = 1 or 0 then we can not comment on ‘m’ and ‘n’.
Case 2: if a = -1, then we can say either both m and n are even or they are odd.
Case 3: in other cases we can say that m = n.

2. If given that (_am) = (_bm)

Case 1: if m = 0, then we can not comment on ‘a’ and ‘b’.
Case 2: if m is odd, then a = b.
Case 3: if m is even, then a = b or a = -b.

SURDS

The numbers of the type ‘ a + √b’ are called surds,
Where ‘a’ and ‘b’ are rational numbers.
If there is a surd of the form a + √b, then a - √b is called the conjugate of this surd.
The product of a surd and its conjugate is always a rational number.

Rationalization of a Surd
When we have a surd in the denominator of any calculation, then it is difficult to perform certain operations on that number. It is always good to have the rational number in the denominator.
So, in order to remove a surd we use a technique of rationalization in which of we have a surd of the form a + √b in the denominator, then we multiply and divide that number with the conjugate of that surd.
Doing this we can remove a surd from the denominator.

Comparison of surds and indices
Surds and indices can be compared among themselves in various ways.
1. By making the bases same.
2. By making the powers same.
3. In case of surds, they can be compared by squaring them.
4. Or surds can also be compared using rationalization method.

Example: Find which one is greater.
1. 299 or 834 834 = 2102 So 834 is greater.
2. 375 or 750
375 = 33×25 = 2725
750 = 72×25= 4925 So, 750is greater.

3. (√7 + √19) or (√5 + √21)
Squaring both of them, we get,
(√7 + √19)2 = 7 + 19 + 2√133 = 26 + 2√133
(√5 + √21)2 = 26 + 2√105 So, now we can say (√7 + √19) is greater.

4. (√23 - √17) or (√29 - √23) Rationalizing both of them,
(√23 - √17) = 6 / (√23 + √17)
(√29 - √23) = 6 / (√29 + √23)
Clearly, (√23 - √17) is greater.