Mathematics Revision: Squares, Cubes, Indices, Factorisation, and Congruence

Flashcards covering the fundamental concepts of square roots, cube roots, powers and indices, algebraic factorisation, and triangle congruence rules based on the revision lecture notes.

Perfect Square Number

A natural number $m$ that can be expressed as $n^2$, where $n$ is also a natural number.

Ending Digits of Square Numbers

Every square number ends with either of the digits $0$, $1$, $4$, $5$, $6$, or $9$ in the unit place.

Ending Zeros of squares

A square number can only possess an even number of zeros at the end, such as $10^2 = 100$ or $400^2 = 160000$.

Relationship between consecutive squares

Between the squares of $n$ and $(n + 1)$, there remain $2n$ natural numbers.

Positive Square Root Symbol

The symbol $\sqrt{}$ is used to denote only the positive square root of a number, for example $\sqrt{49} = 7$.

Cube of a Number

The number obtained when a number is multiplied with itself three times, expressed as $a \times a \times a = a^3$.

Perfect Cube

A number is a perfect cube if, on prime factorisation, each factor occurs exactly three times.

Cube Root Notation

The symbol used to denote a cube root, where if $b^3 = a$, then $\sqrt[3]{a} = b$.

Base and Power

In an expression like $10^4$, $10$ is referred to as the base and $4$ is known as the power or index.

Index Law for Multiplication

For non-zero integers $a$, $m$, and $n$, the law is expressed as $a^m \times a^n = a^{m+n}$.

Power of Zero Law

Any non-zero integer raised to the power of zero is equal to one, expressed as $a^0 = 1$.

Standard Form

A method of expressing large or small numbers as $K \times 10^m$, where $1 \le K < 10$ and $m$ is an integer.

Prime Factorisation

The expression of a number as the product of its factors where every factor involved is a prime number.

Factorisation of Algebraic Terms

The process of expressing an algebraic term in terms of its factors, often using the distributive property for polynomials.

Algebraic Identity for Difference of Squares

The formula used for factorisation: $a^2 - b^2 = (a+b)(a-b)$.

SAS Congruence Rule

Two triangles are congruent if two sides and the included angle of one triangle are equal to the two sides and the included angle of the other triangle.

ASA Congruence Rule

Two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of the other triangle.

SSS Congruence Rule

Two triangles are congruent if the three sides of one triangle are equal to the three sides of another triangle.

RHS Congruence Rule

Two right triangles are congruent if the hypotenuse and one side of one triangle are equal to the hypotenuse and one side of the other triangle.

AAS Congruence Rule

A rule stating two triangles are congruent if they have two equal angles and a corresponding side that is not included between those angles.