Tag Archives: Exponent

Algebraic Factorisation with Exponents (Indices)

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 $2^{n+3} + 16$. \( \begin{align} \displaystyle&= 2^{n+3} + […]

Algebraic Expansion with Exponents (Indices)

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: $$ \large \begin{align} \displaystylea(a+b) &= ab+ac \\(a+b)(c+d) &= ac+ad+bc+bd \\(a+b)(a-b) &= a^2-b^2 \\(a+b)^2 &= a^2 + 2ab + b^2 \\(a-b)^2 &= a^2-2ab […]

Complicated Exponent Laws (Index Laws)

Complicated Exponent Laws (Index Laws)

So far, we have considered situations where one particular exponent’s law was used for simplifying expressions with exponents (indices). However, in most practical situations, more than one law is needed to simplify the expression. The following example simplifies expressions with exponents (indices) using several exponent laws. Example 1 Write $64^{\frac{2}{3}}$ in simplest form. \( \begin{align} […]

Negative Exponents (Negative Indices)

Negative Exponents (Negative Indices)

Consider the following division:$$ \large \dfrac{3^2}{3^3} = 3^{2-3} = 3^{-1}$$Now, if we attempt to calculate the value of this division:$$ \large \dfrac{3^2}{3^3} = \dfrac{9}{27} = \dfrac{1}{3}$$From this conclusion, we can say that $3^{-1} = \dfrac{1}{3}$.This conclusion can be generalised:$$ \large a^{-1} = \dfrac{1}{a}$$ Example 1 Write $4^{-1}$ in fractional form. $4^{-1} = \dfrac{1}{4}$ Example 2 […]