# Laws of Indices, Exponents

Indices are a convenient tool in mathematics to compactly denote the process of taking a power or a root of a number. Taking a power is simply a case of repeated multiplication of a number with itself while taking a root is just equivalent to taking a fractional power of the number. Therefore, it is important to clearly understand the concept as well as the laws of indices to be able to apply them later in important applications.

We will first understand the formal notation for writing a number with an index, followed by the laws governing it. So let’s begin!

## Exponents (or Index Numbers)

### Notation

If we can write a number in the following form –

$$y = a^x$$

The number y then is said to be equal to the number a raised to the power x. a is known as the base (the number which is to be multiplied by itself successively) and x is known as the power or index to which is raised, or simply the exponent of a.
It can also be equivalently interpreted as –

$$a^x = a\times a \times a$$ …… (x times)

can be any real number. a can be any real number for x ∈ Ζ (Integer) and is restricted to being a positive real number for fractional values of x.

## Laws of Indices

For real numbers m,n and valid bases a,b, the following basic laws hold –

### Law 1

$$a^m \times a^n = a^{(m + n)}$$
Note that for this law to be applicable, the bases of both of the numbers to be multiplied must be the same.

### Law 2

$$\frac{a^m}{a^n} = a^{(m – n)}$$

#### Important Result –

For applying the above Law, if we choose both m = 1 and n = 1, then we get –

$$\frac{a^1}{a^1} = a^{(1 – 1)}$$

$$\frac{a}{a} = a^0$$

$$a^0 = 1$$

This is a very important result and we’ll use it often to simplify our algebraic expressions involving indices.

### Law 3

$$(a^m)^n = a^{mn} = (a^n)^m$$

#### Example –

$$(a^{\frac{1}{2}})^3 = (\sqrt{a})^3 = (a^3)^{\frac{1}{2}}$$ $$= a^{\frac{3}{2}}$$

### Law 4

$$(ab)^n = a^nb^n$$

The four laws mentioned above are sufficient for evaluating any arbitrary expression involving indices.
The solved examples below will further clear your doubts if any.

## Solved Examples on Laws of Indices, Exponents

Question 1: Show that for any positive real number p, the expression $$a^{-p}$$ is equivalent to $$\frac{1}{a^p}$$.

Solution: We proceed with the following manipulation –

$$a^{-p} = a^{(0 – p)}$$

Using Law 2 i.e. $$\frac{a^m}{a^n} = a^{(m – n)}$$, we can rewrite the above expression as –

$$\frac{a^0}{a^p}$$ $$= \frac{1}{a^p}\text{ , which is the required result.}$$

Note ⇒ Using this result, we can use the Law 1 of Indices to derive the Law 2 as well. Just substitute n as -n and see for yourself!
Besides, logically also, taking a negative exponent should mean division, because the inverse function of multiplication is indeed division.

Question 2: Simplify and evaluate $$(\frac{16}{81})^{-\frac{3}{4}}$$

Solution: Using the laws of indices and some manipulation –
$$(\frac{16}{81})^{-\frac{3}{4}} = \frac{1}{(\frac{16}{81})^{\frac{3}{4}}}$$$$= (\frac{81}{16})^{\frac{3}{4}}$$$$= ( (\frac{81}{16})^{\frac{1}{4}})^3$$$$= (\frac{81^{\frac{1}{4}}}{16^{\frac{1}{4}}})^3$$$$=(\frac{3}{2})^3$$$$= \frac{3^3}{2^3}$$$$= \frac{27}{8}$$

Question 3: Simplify the expression $$y = x^{a – b} \times x^{b – c} \times x^{c – a} \times x^{-a-b}$$

Solution: Using Law 1 –

$$y = x^{a – b} \times x^{b – c} \times x^{c – a} \times x^{-a-b}$$

⇒ $$y = x^{(a – b) + (b – c) + (c – a) + (-a – b)}$$

⇒ $$y = x^{-a -b}$$

$$y = \frac{1}{x^{a + b}}$$

This is the final simplified expression.

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