Surds and Indices

Exponents and Powers

Very large numbers are difficult to read, understand, compare and operate upon. To make all these easier, you can use exponents, converting many of the large numbers in a shorter form.

For example,

  • 10,000 = 104 (read as 10 raised to 4)
  • 243 = 35
  • 128 = 27

Here, 10, 3 and 2 are the bases, whereas 4, 5 and 7 are their respective exponents. You can also say, 10,000 is the 4th power of 10, 243 is the 5th power of 3, etc.

Very small numbers can be expressed in standard form using negative exponents.

Three Laws of Indices

There are just 3 laws and from those, you can derive another 3 other interesting rules.

1. xm.xn=x(m+n) 

2. xm/xn=x(m-n) 

3. (xm)n=xmn

Three Additional Rules of Indices

4. Anything to the power zero is 1. Put n=m in the above second law, you will get this rule.

x= 1

5. Rule of negative powers; Put n=-m in the first law above to get this rule.

x-m = 1/xm

6. Rule of fractional powers; Put n=1/m in the third law above to get this rule.

(xm)1/m = x

Other Rules

7. xm x ym = (xy)m

8. xm/ym = (x/y)m


If n is a positive integer and a is a positive rational number (a > 0), then n√a is called a surd of order n.

Simple Surd: A surd which consists of a single term is called surd or monomial surd.

Mixed Surd: If a is a rational number and √b is a surd, then a + √b, a - √b are called mixed surds.

Compound Surd: A surd which is a sum or difference of two or more surds is called a compound surd.

Similar Surds: If two surds are different multiples of the same surd, they are called similar surds otherwise they are dissimilar surds.

Rationalization of a Surd

If the product of two surds is rational, then each of the two surds is called a rationalizing factor of the other. In general, if the surd is of type a + √b, then its rationalizing factor a - √b.