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GRE Math Book. It provides a cuttingedge, indepth overview of all the math concepts from basic to midupper levels. The book still remains our hallmark: from basic to the most advanced GRE math concepts tested during the exam. Moreover, the following chapters will give you many tips, tricks, and shortcuts to make your quant preparation more robust and solid.
Source for some of the images in this post:
A Maths Dictionary for Kids by © Jenny Eather
1) Exponents
Basic operations of indices 
Math operations
 Examples

\(a^m*a^n=a^{m+n}\)  \(7^3*7^5=7^{3+5}=7^8\) 
\(\frac{a^m}{a^n}=a^{mn}\)  \(\frac{7^3}{7^5}=7^{35}=2^{2}\) 
\((a^m)^n=a^{m*n}\)  \((7^2)^3=7^{2*3}=7^6\) 
\(x^{n}=\frac{1}{x^n}\)  \(7^{2}=\frac{1}{7^2}\) 
\((\frac{a}{b})^{n}=(\frac{b}{a})^n\)  \((\frac{1}{7})^{2}=7^2\) 
\(\frac{1}{a^m}=a^{m}\)  \(\frac{1}{7^3}=7^{3}\) 
\(\sqrt[n]{a^m}=(a^m)^{\frac{1}{n}}=a^{\frac{m}{n}}\)  \( \sqrt[3]{7^6}\)\(=(7^6)^{\frac{1}{3}}=7^{\frac{6}{3}}=7^2=49\) 
\(a^0=1\), if \(a \neq 0\)  \(7^0=1\) 
\((a)^{even}=+(a)^{even}\) \(a>0\)  \((7)^4=7 \times 7 \times 7 \times 7 = 7^4=2,401\) 
\((a)^{odd}=(a)^{odd}\) \(a>0\)  \((7)^3=(7 \times 7 \times 7)=343\) 
Note \((a^m)^n \neq a^{m^n}\)
\((7^2)^3 = 7^6\)
However
\(7^{3^2}=7^9\). You have to elevate \(3^2\) before and then \(7^9\)
\(7^6 \neq 7^9
\)
2) Unit Digit Power
Unit Digit or Last Digit of a^n 
a\n
 1
 2
 3
 4
 cyclicity

0  0  0  0  0  1 
1  1  1  1  1  1 
2  2  4  8  6  4 
3  3  9  7  1  4 
4  4  6  4  6  2 
5  5  5  5  5  1 
6  6  6  6  6  1 
7  7  9  3  1  4 
8  8  4  2  6  4 
9  9  1  9  1  2 
ExampleFind the unit digit of \(7^{99}\)?
Having a cyclicity of 4 we do have that \(7^{10}\) has a unit digit of 9. This for nine times which is \(7^{90} =\) unit digit of 9 + 9 spots the unit digit is
33) Unit Digit of a SquareThe square of a number can never end with \(2, 3, 7, 8\) or odd number of zeros.4) Number rearrangementWhen a twodigit number is reversed then the
sum of two numbers is
always divisible by 11 & the
difference of two numbers is
always divisible by 9Let's say we have a number
92When reversed it will be
29Sum = 92+29 =
\(121\) > Divisible by
\(11\)Difference = 9229 =
\(63\) > Divisible by
\(9\) 5) Square of 5Suppose that X is a number ending in 5 or 5 itself
\((X5)^2=X(X+1),25\)
\(625^2=62 \times 63\),
\(25\)\(=\)
\(3906\)\(25\)You just need to multiply \(62 \times 63\) and add
\(25\)6) Square of any number\(n^2=(nd)(n+d)+d^2\)
d is the distance from the
nearest multiple of
10.100\(19^2=(191)(19+1)+1^2=18 \times 20+1=361\)
2) Roots1) Properties of radicals2) Surds• another name for an irrational number.
• a surd is a real number that can be written as a nonrepeating or nonterminating decimal but not as a fraction because the decimal goes on forever without repeating.
source:
http://www.amathsdictionaryforkids.com/Surds are irrational roots of a rational number. e.g. √6 = a surd ⇒ it can’t be exactly found. Similarly – √7, √8, \(\sqrt[3]{9}\), \(\sqrt[4]{27}\) etc. are all surds.
 Pure Surd : The surds which are made up of only an irrational number e.g. √6, √7, √8 etc.
 Mixed Surd : Surds which are made up of partly rational and partly irrational numbers e.g. 3√3, \(6^4√27\) etc.
 Rationalization of Surds: In order to rationalize a given surd, multiply and divide by the conjugate of denominator [conjugate of (a + √b) is (a – √b) and vice versa].
e.g. \(\frac{(6+\sqrt{2})}{(1\sqrt{3})}=\frac{(6+\sqrt{2})(1+\sqrt{3})}{(1\sqrt{3})(1+\sqrt{3})}=\frac{(6+6 \sqrt{3}+\sqrt{2}+\sqrt{6})}{(13)}=\frac{(6+6 \sqrt{3} +\sqrt{2}+\sqrt{6})}{2}\)
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