# 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 […]