# Exponential Equations (Indicial Equations)

Home > Algebra The equation $a^x=y$ is an example of a general exponent equation (indicial equation) and $2^x = 32$ is an example of a more specific exponential equation (indicial equation). To solve one of these equations it is necessary to write both sides of the equation with the same base if the unknown is [...]

# 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 [...]

# Multiplication using Exponents (Indices)

Home > Algebra If we wish to calculate $5^4 \times 5^3$, we could write in factor form to get: \begin{align} \displaystyle 5^4 \times 5^3 &= (5 \times 5 \times 5 \times 5) \times (5 \times 5 \times 5) \\ &= 5^7 \\ \end{align} Example 1 Simplify $7^2 \times 7^3$ after first writing in [...]

$\textbf{Introduction to Proof by Contradiction}$ 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 […]

# The Sign of One

Make the following numbers using FOUR 1s using any mathematics operators and/or symbols, such as $\dfrac{x}{y}$, $\sqrt{x}$, decimal dots, $+$, $-$, $\times$, $\div$, $($ $)$, etc by the Sign of One. Click the numbers below to see the answers. The first one is done for you.

# Number Curiosity

Are these numbers beautiful? \begin{align} \displaystyle 1 \times 8 + 1 &= 9 \\ 12 \times 8 + 2 &= 98 \\ 123 \times 8 + 3 &= 987 \\ 1234 \times 8 + 4 &= 9876 \\ 12345 \times 8 + 5 &= 98765 \\ 123456 \times 8 + 6 &= 987654 \\ 1234567 […]

# 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 […]

# 10 Deadly Common Algebra Mistakes

Share0 Share +10 Tweet0 Common Patterns of Algebra Mistakes Students often make Common Algebra Mistakes due to confusions such as expand and simplify rules, fractions, indices and equations which that lead the students to the wrong answer. Also these error patterns are very basic and quite easily rectified. Check yourself whether you make similar mistakes, […]

# Absolute Value Inequalities

Share0 Share +10 Tweet0 Absolute Value Inequalities are usually proved by the absolute value of a certain value is greater than or equal to it. The square of the value is equal to the square of its absolute value. Proof of Absolute Value Inequalities Prove $|a| + |b| \ge |a+b|$.