Monday, August 7, 2017

97-GRE Math Test - Important Key points and formulas Part - 2

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 and read as "5 to the power 4" or "5 raised to the 4th power" 5 = 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
38) In the expression, 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, 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,  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) m   x  n  m + n
42) m    n  m - n
43) (m n  m x n
44) o  =  1
45) 1  =  a
46) -m  =  1/m    ,  -1  =  1/a ) , ( 1/-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, xyz3 + x2yz3 there are two terms which are separated by + sign whereas in x4y3z7it 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 + ab
Here 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.