We learnt that the simplest exponential functions are $y=a^x$ where $a>0$, $a \ne 1$.For all positive values of the base $a$, the graph is always positive, that is $a^x > 0$ for all $a>0$.There are an infinite number of possible choices for the base number. However, where exponential data is examined in engineering, science, and […]

# Tag Archives: Exponent

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

# 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)

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} […]

# Rational Exponents (Rational Indices)

$\textit{Square Root}$ Until now, the exponents (indices) have all been integers. In theory, an exponent (index) can be any number. We will confine ourselves to the case of exponents (indices), which are rational numbers (fractions).The symbol $\sqrt{x}$ means the square root of $x$. It means finding a number that multiplies to give the original $x$. […]

# 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 […]

# Raising a Power to Another Power

If we are given $(2^3)^4$, that can be written in factor form as $2^3 \times 2^3 \times 2^3 \times 2^3$.We can then simplify the multiplication using the exponent’s rule as $2^{3+3+3+3} = 2^{12}$. Similarly, if we are given $(5^2)^3$, this means;\( \begin{align}(5^2)^3 &= 5^2 \times 5^2 \times 5^2 \\&= 5^{2+2+2} \\&= 5^6 \end{align} \) Using […]

# Division using Exponents (Indices)

If we are given $a^8 \div a^3$, we can also write this as $\dfrac{a^8}{a^3}$, which means $\dfrac{a \times a \times a \times a \times a \times a \times a \times a}{a \times a \times a}$.As there are \(8\) factors of $a$ on the top line (numerator), and $3$ factors of $a$ on the bottom line […]

# Multiplication using Exponents (Indices)

If we wish to calculate $5^4 \times 5^3$, we could write in factor form to get:\( \begin{align} \displaystyle5^4 \times 5^3 &= (5 \times 5 \times 5 \times 5) \times (5 \times 5 \times 5) \\&= 5^7 \end{align} \) Example 1 Simplify $7^2 \times 7^3$ after first writing in factor form. \( \begin{align} \displaystyle7^2 \times 7^3 […]