In continuation of part - 1, we will see all the important formulas and useful statements which are to be used in the Math test 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 the "m-th power of n-th root of a" or we can also say that it is the "n-th root of m-th power of a". It is also denoted as n√ am
38) 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, a1/n, here, it is the n-th root of "a". For any even value of "n", there are exactly two roots for every positive number "a", and no roots if "a" is negative.
39) In the expression, a1/n, here, it is the n-th root of "a". For any even value of "n", there are exactly two roots 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.
42) a m / a n = a m - n
43) (a m ) n = a m x n
44) a o = 1
45) a 1 = a
46) a -m = 1/a m , ( a -1 = 1/a ) , ( 1/a -1 = a )
Rules of Indices
41) a m x a n = a m + n42) a m / a n = a m - n
43) (a m ) n = a m x n
44) a o = 1
45) a 1 = a
46) 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 a 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 a + or - sign.
In an expression, xyz3 + x2yz3 there are two terms that are separated by the + sign whereas in x4y3z7it is the single term.
= 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 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)
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 a few examples and some essential formulae.
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