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
\(a^m*a^n=a^{m+n}\)	\(7^3*7^5=7^{3+5}=7^8\)
\(\frac{a^m}{a^n}=a^{m-n}\)	\(\frac{7^3}{7^5}=7^{3-5}=2^{-2}\)
\((a^m)^n=a^{m*n}\)	\((7^2)^3=7^{2*3}=7^6\)
\(x^{-n}=\frac{1}{x^n}\)	\(7^{-2}=\frac{1}{7^2}\)
\((\frac{a}{b})^{-n}=(\frac{b}{a})^n\)	\((\frac{1}{7})^{-2}=7^2\)
\(\frac{1}{a^m}=a^{-m}\)	\(\frac{1}{7^3}=7^{-3}\)
\(\sqrt[n]{a^m}=(a^m)^{\frac{1}{n}}=a^{\frac{m}{n}}\)	\( \sqrt[3]{7^6}\)\(=(7^6)^{\frac{1}{3}}=7^{\frac{6}{3}}=7^2=49\)
\(a^0=1\), 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





Example

Find 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 3



3) Unit Digit of a Square

The square of a number can never end with \(2, 3, 7, 8\) or odd number of zeros.



4) Number rearrangement


When a two-digit number is reversed then the sum of two numbers is always divisible by 11 & the difference of two numbers is always divisible by 9

Let's say we have a number 92

When reversed it will be 29
Sum = 92+29 = \(121\) ----> Divisible by \(11\)
Difference = 92-29 = \(63\) ----> Divisible by \(9\)



5) Square of 5

Suppose 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

\(n^2=(n-d)(n+d)+d^2\)

d is the distance from the nearest multiple of 10.100

\(19^2=(19-1)(19+1)+1^2=18 \times 20+1=361\)



2) Roots


1) Properties of radicals





2) 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. \(\frac{(6+\sqrt{2})}{(1-\sqrt{3})}=\frac{(6+\sqrt{2})(1+\sqrt{3})}{(1-\sqrt{3})(1+\sqrt{3})}=\frac{(6+6 \sqrt{3}+\sqrt{2}+\sqrt{6})}{(1-3)}=\frac{(6+6 \sqrt{3} +\sqrt{2}+\sqrt{6})}{-2}\)