Algebraic Factorisation with Exponents (Indices)

Algebraic Factorisation with Exponents (Indices)

Home > Algebra $\textit{Factorisation}$ We first look for $\textit{common factors}$ and then for other forms such as $\textit{perfect squares}$, $\textit{difference of two squares}$, etc. Example 1 Factorise $2^{n+4} + 2^{n+1}$. Show Solution \( \begin{align} \displaystyle &= 2^{n+1} \times 2^{3} + 2^{n+1} \\ &= 2^{n+1}(2^{3} + 1) \\ &= 2^{n+1} \times 9 \\ \end{align} \) Example [...]
Algebraic Expansion with Exponents (Indices)

Algebraic Expansion with Exponents (Indices)

Home > Algebra $\textit{Algebraic Expansion with Exponents}$ Expansion of algebraic expressions like $x^{\frac{1}{3}}(4x^{\frac{4}{5}} - 3x^{\frac{3}{2}})$, $(4x^5 + 6)(5^x - 7)$ and $(4^x + 7)^2$ are handled in the same way, using the same expansion laws to simplify expressions containing exponents: $$ \begin{align} \displaystyle a(a+b) &= ab+ac \\ (a+b)(c+d) &= ac+ad+bc+bd \\ (a+b)(a-b) &= a^2 -b^2 [...]
Perfect Numbers

Perfect Numbers

Some of the first patterns of natural numbers spotted and studied by Pythagoreans were $\textit{perfect numbers}$. These are natural numbers with a bizarre property that they can be formed by adding up all the smaller numbers that make up their divisors. The number $6$ is the first perfect number, because it may be divided by […]

Method of Exhaustion

Method of Exhaustion

Eudoxus, Greek mathematician 408-355 BC, developed the idea of seeking mathematical solutions using he $\textit{Method on Exhaustion}$: at this stage, you achieve a more accurate figure. To find an increasingly accurate solution to $\sqrt{2}$, for example he produced a ladder of numbers. Starting with $1$ and $1$ as the first row, he then added those […]

The Golden Ratio

The Golden Ratio

Since the days of old, artists as well as mathematicians have known that there is a special, aesthetically pleasing rectangle with width $1$, length $x$, and the following property: When a square of side $1$ is removed, the rectangle that remains has the same proportions as the original rectangle. Since the new rectangle has a […]

Proof by Contradiction

Proof by Contradiction

$\textbf{Introduction to Proof by Contradiction}$ The basic idea of $\textit{Proof by Contradiction}$ is to assume that the statement that we want to prove is $\textit{false}$, and then show this assumption leads to nonsense. We then conclude that it was wrong to assume the statement was $\textit{false}$, so the statement must be $\textit{true}$. As an example […]

Nicomachus Theorem

Nicomachus Theorem

$\textbf{Nicomachus}$ discovered “Nicomachus Theorem” interesting number patterns involving cubes and sums of odd numbers. Nicomachus was born in Roman Syria (now, Jerash, Jordan) around 100 AD. He wrote in Greek was a Pythagorean. $\textbf{Nicomachus Theorem: Cubes and Sums of Odd numbers}$ \begin{eqnarray*} 1 &=& 1^3 \\ 3 + 5 &=& 8 = 2^3 \\ 7 […]