Laws of Indices and Rational Exponents

Defining the Notation

Indices, also known as powers or exponents, are a concise method for writing repeated multiplication. We start with the basic definition for a positive whole number $m$: the expression $a^m$ represents the number $a$ multiplied by itself a total of $m$ times.

$a^m = a \times a \times a \times \dots \times a$ ($m$ factors of $a$)

This notation allows us to derive our first rule. If we multiply $a^m$ by $a^n$, we are multiplying a string of $m$ factors of $a$ by a string of $n$ factors of $a$. The result is a single string of $m+n$ factors of $a$:

$a^m \times a^n = \underbrace{(a \times \dots \times a)}_{m \text{ times}} \times \underbrace{(a \times \dots \times a)}_{n \text{ times}} = a^{m+n}$

RULE 1: $a^m \times a^n \equiv a^{m+n}$

While this rule is derived using positive integers, we decide to extend it to all real numbers to ensure our notation remains consistent across mathematics.

Rational Exponents and Roots

To understand what a fractional power like $a^{\frac{1}{3}}$ means, we apply Rule 1. If we multiply $a^{\frac{1}{3}}$ by itself three times, we must add the exponents:

$a^{\frac{1}{3}} a^{\frac{1}{3}} a^{\frac{1}{3}} = a^{(\frac{1}{3} + \frac{1}{3} + \frac{1}{3})} = a^1 = a$

Since $a^{\frac{1}{3}}$ multiplied by itself three times equals $a$, it must be interpreted as the cube root of $a$. This leads to our second rule for any positive integer $n$:

RULE 2: $a^{\frac{1}{n}} = \sqrt[n]{a}$

Negative Indices and the Zero Power

We can determine the meaning of negative indices by exploring how Rule 1 applies to an expression like $a^3 \times a^{-2}$. According to the rule:

$a^3 \times a^{-2} = a^{3+(-2)} = a^1 = a$

To get $a$ from $a^3$, we must multiply by $\frac{1}{a^2}$. This leads to the definition of negative powers:

RULE 3: $a^{-m} = \frac{1}{a^m}$

Similarly, we can derive the value of $a^0$ by considering $a^2 \times a^{-2}$:

$a^2 \times a^{-2} = a^{2-2} = a^0$

Alternatively, using Rule 3:

$a^2 \times a^{-2} = a^2 \times \frac{1}{a^2} = \frac{a^2}{a^2} = 1$

For consistency, we must define the zero power as follows:

RULE 4: $a^0 = 1$ (for $a > 0$)

Further Laws and General Rational Exponents

Building on these foundations, we can establish the remaining laws of indices that are essential for the TMUA and ESAT:

RULE 5 (Division): $a^m \div a^n = \frac{a^m}{a^n} = a^m \times a^{-n} = a^{m-n}$

RULE 6 (Powers of Powers): $(a^m)^n = a^{mn}$

RULE 7 (General Rational Exponent): $a^{\frac{m}{n}} = (a^m)^{\frac{1}{n}} = \sqrt[n]{a^m} = (a^{\frac{1}{n}})^m = (\sqrt[n]{a})^m$

When applying Rule 6, you must be extremely careful with notation. The expressions $(a^m)^n$ and $a^{m^n}$ are not equivalent. For example:

$(a^3)^2 = a^{3 \times 2} = a^6$, whereas $a^{3^2} = a^9$.

Why Bases Must Be Positive

The index laws are generally applied where $a$ is a positive number. If the base $a$ is negative, rational powers can become problematic. For instance, $(-64)^{\frac{1}{3}} = -4$ is well defined in real numbers, but $(-64)^{\frac{1}{2}}$ would require the square root of a negative number, which does not exist in the real number system. To avoid these inconsistencies, the specification focuses on positive $a$.

Irrational Exponents

Although the specification focuses on rational exponents, these rules actually apply to irrational numbers as well. One way to visualise $2^{\sqrt{3}}$ is to consider the graph of $y = 2^x$. While calculating $2^x$ for rational $x$ gives a series of discrete points, these points are so close together that they form the appearance of a curve. The values for irrational $x$ are those that precisely fill the gaps to create a continuous, unbroken curve. Mathematically, these are found using the concept of limits, ensuring that the value of $2^{\sqrt{3}}$ lies between $2^p$ and $2^q$, where $p$ and $q$ are rational numbers slightly smaller and larger than $\sqrt{3}$ respectively.