Exploring the depths of indices calculations unveils a fascinating aspect of algebra essential for mastering IGCSE mathematics. This section, crafted with a focus on calculations involving various indices including those with negative and positive powers, is designed to enhance your understanding through a blend of theory and practical examples.

**Understanding Indices**

Indices, or powers, allow us to write repeated multiplication compactly. A power, $a^n$, consists of a base (a) and an exponent (n), indicating how many times the base is multiplied by itself. This segment ventures into the intricacies of performing calculations with these powers, especially focusing on manipulating expressions with both negative and positive exponents.

Image courtesy of OnlineMathLearning

**Positive and Negative Powers**

**Positive powers**signify the base multiplied by itself $n$ times.**Negative powers**represent the reciprocal of the base raised to the positive power. For instance, $a^{-n} = \frac{1}{a^n}$.

**Zero Power**

A crucial principle in indices is that any base (except zero) raised to the power of zero equals one, $a^0 = 1$, symbolizing the absence of multiplication.

**Calculations with Indices**

Let's delve into the specifics of performing various calculations with indices, elucidating each concept with examples.

Image courtesy of Cuemath

**Multiplication of Powers**

To multiply powers with the same base, add their exponents:

$a^n \times a^m = a^{n+m}$**Division of Powers**

To divide powers with the same base, subtract the exponents:

$a^n \div a^m = a^{n-m}$**Raising a Power to a Power**

To raise a power to another power, multiply the exponents:

$(a^n)^m = a^{n \times m}$**Working with Fractions**

When a fraction is raised to a power, both the numerator and the denominator are raised to that power:

$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$## Worked Examples

### Example 1:

Simplify $3^4 \times 3^{-1}$.

#### Solution:

$3^4 \times 3^{-1}$

$= 3^{4-1}$

$= 3^3$

$= 27$

**Example 2:**

Calculate $\left(\frac{4}{5}\right)^{-2}$.

**Solution:**

$(\frac{4}{5})^{-2}$

$= (\frac{5}{4})^2$

$= \frac{5^2}{4^2}$

$= \frac{25}{16}$

$= 1.5625$

### Example 3:

Simplify $(2^3)^{-2}$.

**Solution:**

$(2^3)^{-2}$

$= 2^{3 \times -2}$

$= 2^{-6}$

$= \frac{1}{2^6}$

$= \frac{1}{64}$

$= 0.015625$