We learnt that the simplest exponential functions are of the form $y=a^x$ where $a>0$, $a \ne 1$. We can see that 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, […]

# 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 $2^{n+3} + 16$. […]

# 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 &= […]

# 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 shows a simplification of expressions with exponents (indices), using several exponent laws. Example 1 Write $64^{\frac{2}{3}}$ in simplest […]

# 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 the rational number (fractions). The symbol $\sqrt{x}$ means square root of $x$. It means, finds a number that multiplies by itself to give […]

# Negative Exponents (Negative Indices)

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

# 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 […]

# Multiplication using Exponents (Indices)

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 factor form. \( […]