Indices: Master the Laws of Indices for O Level Additional Math

The laws of indices are a set of techniques to be applied when simplifying expressions containing powers. The purpose of the laws is to enable us to simplify problems of addition, subtraction, multiplication, and division involving powers. They are useful in many branches of mathematics, both for reducing lengthy calculations and for allowing us to work out a solution by inspection. There are $5$ important laws of indices.

In general, if $a$ is real number, and $m$ and $n$ are positive integers, then

Law 1 of Indices (same base): ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$

Example: ${{6}^{4}}\times {{6}^{5}}={{6}^{9}}$

Law 2 of Indices (same base): $\frac{{a}^{m}}{{a}^{n}}={{a}^{m-n}}$, if $a \neq 0$.

Example: $\frac{{{10}^{6}}}{{{10}^{4}}}={{10}^{2}}$

Law 3 of Indices (same base): ${{({{a}^{m}})}^{n}}={{a}^{mn}}$

Example: ${{\left( {{2}^{5}} \right)}^{2}}={{2}^{10}}$

If $a$ and $b$ are real numbers, and $n$ is a positive integer, then

Law 4 of Indices (same index): ${{a}^{n}}\times {{b}^{n}}={{\left( ab \right)}^{n}}$

Example: ${{\left( -3 \right)}^{2}}\times {{\left( -4 \right)}^{2}}={{\left( -3\times \left( -4 \right) \right)}^{2}}$

Law 5 of Indices (same index): $\frac{{{a}^{m}}}{{{b}^{m}}}={{\left( \frac{a}{b} \right)}^{m}}$

Example: $\frac{{{8}^{3}}}{{{5}^{3}}}={{\left( \frac{8}{5} \right)}^{3}}$

The negative index notation is a convenient way to deal with situations where the power is a number less than $0$. What happens if we have a number with a negative power? Generally, we define it as

${a}^{-n}=\frac{1}{a^n}$

where $a$ is any real number and $a\neq 0$.

To simplify $\left( \frac{{{a}^{4}}}{{{a}^{6}}} \right)$,

We can use Law 2 of Indices, $\left( \frac{{{a}^{4}}}{{{a}^{6}}} \right)={a}^{4-6}={a}^{-2}$

But we also know that $\left( \frac{{{a}^{4}}}{{{a}^{6}}} \right)=\frac{1}{{a}^{2}}$

Therefore, ${a}^{-2}=\frac{1}{{a}^{2}}$

From this, we can derive: ${{\left( \frac{a}{b} \right)}^{-n}}={{\left( \frac{b}{a} \right)}^{n}}$

We can now extend all the $5$ laws of indices to include all integer indices (positive, zero, and negative).

Since we have already extended all the 5 laws of indices to include all integer indices, is it possible to have an index that is not an integer? What happens if we encounter indices that are non-integer rational numbers?

If $a$ is a positive integer, we can define that

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

if $a>0$.

Examples: ${7}^{\frac{1}{5}}=\sqrt[5]{7}$ ${10}^{\frac{1}{11}}=\sqrt[11]{10}$

Generally, if $m$ and $n$ are positive integers,

${{a}^{\frac{m}{n}}}=\sqrt[n]{{{a}^{m}}}$ or ${{\left( \sqrt[n]{a} \right)}^{m}}$

if $a>0$.

Suppose you have to solve a complicated question with exponential equations, you can often make the question simpler by a process called substitution. The idea is to replace a hard-to-solve equation with an easier one.

It is called substitution because what gets substituted is a simpler expression that can then be solved using more basic math skills. You can substitute any expression for any variable in an equation as long as you don't change the value of whatever expression you're solving for.