In continuation of

**Blog-96**, we will see all the important formulas and useful statements which are to be used in Math test of GRE.### Exponents and Roots:

36) When the same numbers are multiplied several times then the concept of exponents will be used. 5 x 5 x 5 x 5, here we multiplied 5, 4 times so we call 5 as the base and the number 4 is called an exponent. It is written as 5

^{4 }and read as "**5 to the power 4**" or "**5 raised to the 4th power**" 5^{4 }= 5 x 5 x 5 x 5 = 625.
37) a

^{m/n }is "**a to the power m/n**". Here m/n is the power. It means it is "**m-th power of n-th root of a**" or we can also say that it is "**n-th root of m-th power of a**". It is also denoted as^{n}√ a^{m }
38) In the expression,

39) In the expression,

**a**^{1/n }, here, it is the n-th root of "**a**". For any odd value of "**n**", there is exactly one root for every number "**a**", even if "**a**" is negative.39) In the expression,

**a**^{1/n }, here, it is the n-th root of "**a**". For any even value of "**n**", there is exactly two root for every positive number "**a**", and no roots if "**a**" is negative.
40) Example: 27 has exactly one cube root,

42) a

43) (a

44) a

45) a

46) a

^{3}√ 27 = 3, but 27 has two 4th roots,^{4}√ 27 and (-^{4}√ 27 ), and -27 has only one cube root,^{3}√ -27 = -3 but -27 has no 4th root, as it is negative.### Rules of Indices

41) a^{m }x a^{n }= a^{m + n}42) a

^{m }/ a^{n }= a^{m - n}43) (a

^{m })^{n }= a^{m x n}44) a

^{o}^{ }= 145) a

^{1}^{ }= a46) a

^{-m}^{ }= 1/a^{m , }( a^{-1}^{ }= 1/a ) , ( 1/a^{-1}^{ }= a )### Operations with Algebraic Expression

47) One or more variables are available in an algebraic expression. Each expression can be expressed as a single term or sum of two or more terms. Terms of an expression are separated by addition or subtraction. In other words, two or more terms are separated by + or - sign.

In an expression, xyz

###
Factorization (

= 3 x 21 x 2 x 5

= 3 x 3 x 7 x 2 x 5

= 2 x 3 x 3 x 5 x 7 ( write it in ascending order).

= x

= x

= x

Here the factors of the polynomial [x

(x + a) and (x + b)

^{3}+ x^{2}yz^{3}there are two terms which are separated by + sign whereas in x^{4}y^{3}z^{7}it is the single term.###
Factorization (**Factorisation**):

#### a) Factorization of Integers:

#### For any integers, we can find the prime factors. Let us find the prime factors of 630.

630 = 63 x 10= 3 x 21 x 2 x 5

= 3 x 3 x 7 x 2 x 5

= 2 x 3 x 3 x 5 x 7 ( write it in ascending order).

#### b) Factorization of Polynomial:

(x + a) (x + b) = x (x + b) + a (x + b)= x

^{2}+ b x + a x + ab= x

^{2}+ a x + b x + ab= x

^{2}+ (a + b) x + abHere the factors of the polynomial [x

^{2}+ (a + b) x + ab], are(x + a) and (x + b)

In the next part we will see few examples and some important formulae.