# Algebra

# Rounding to the Nearest Whole Numbers

# Inequalities involving Absolute Values and Surds

For handling inequalities involving absolute values and surds, it is required to ensure the domains before solving inequalities. the final solutions must fit in the domains. Example 1 Solve for \( x \), \( |x| \gt \sqrt{x+2} \). \( \begin{align} x+2 &\ge 0 &\color{green}{\text{domain of } \sqrt{x+2} } \\ x &\ge -2 \color{green}{\cdots (1)} \\ […]

# Exponential Equations (Indicial Equations)

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 an exponent (index) […]

# Algebraic Factorisation with Exponents (Indices)

$\textit{Factorisation}$ 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 $2^{n+4} + 2^{n+1}$. \( \begin{align} \displaystyle &= 2^{n+1} \times 2^{3} + 2^{n+1} \\ &= 2^{n+1}(2^{3} + 1) \\ &= 2^{n+1} \times 9 \\ \end{align} \) Example 2 Factorise $2^{n+3} + 16$. […]

# Algebraic Expansion with Exponents (Indices)

$\textit{Algebraic Expansion with Exponents}$ 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 \\ (a+b)^2 &= […]

# 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 […]