Exploring foundational concepts of powers and roots, this section is critical for IGCSE Maths students to grasp the nature of various number types. It simplifies understanding of squares, square roots, cubes, and cube roots, with a focus on recall techniques. The content is designed to be both thorough and succinct, enhancing learning outcomes.

**Powers & Indices**

**Understanding Powers**

Powers or indices denote the operation of multiplying a number by itself a specified number of times.

- $2^1 = 2$
- $2^2 = 4$ (since $2 \times 2 = 4$)
- $2^3 = 8$ (as $2 \times 2 \times 2 = 8$)
- $2^4 = 16$ (as $2 \times 2 \times 2 \times 2 = 16$)
- $2^5 = 32$ (as $2 \times 2 \times 2 \times 2 \times 2 = 32$)
- $2^0 = 1$, for any non-zero base.

**Understanding Roots**

Roots perform the inverse operation of powers, identifying the base number from its powered result.

- Square root: $\sqrt{25} = 5$, indicating both $5$ and $-5$ are roots.
- Cube root: $\sqrt[3]{125} = 5$, unique for both positive and negative numbers.

**Reciprocals**

The reciprocal of a number is what you multiply by to get 1, for example:

- Reciprocal of $2$ is $\frac{1}{2}$.
- $5^{-1}$ signifies the reciprocal of $5$, and $5^{-2}$ the reciprocal of $5^2$.

**Laws of Indices**

These laws facilitate the simplification of expressions and computations.

**Worked Examples**

**Example 1**:

Calculate $3^4$.

**Solution**:

$3^4 = 3 \times 3 \times 3 \times 3 = 81$**Example 2**:

Simplify $\sqrt{49}$.

**Solution**:

$\sqrt{49} = 7$

As $(7 \times 7 = 49)$

**Example 3**:

Evaluate $\left(\frac{2}{3}\right)^{-3}.$