In this section, we delve into the fascinating world of indices calculations, an essential concept in algebra that plays a pivotal role in simplifying and solving mathematical expressions. Our focus is on equipping students with the knowledge and skills to confidently handle indices, enabling them to simplify expressions involving powers. This includes understanding how to manipulate expressions with positive, zero, and negative indices, ensuring a solid foundation in algebra that will be invaluable for their IGCSE exams and beyond.

**Understanding Indices Calculations**

Indices, or powers, are a shorthand way of expressing repeated multiplication of a number by itself. Calculating with indices is a fundamental skill in algebra, allowing us to simplify expressions efficiently. Before diving into complex calculations, it's crucial to grasp the basic laws of indices, which are the cornerstone of this topic.

**The Laws of Indices**

**Multiplication of Powers (Same Base):**$a^m \times a^n = a^{m+n}$**Division of Powers (Same Base):**$a^m \div a^n = a^{m-n}$**Power of a Power:**$(a^m)^n = a^{mn}$**Power of a Product:**$(ab)^n = a^n b^n$**Negative Indices:**$a^{-n} = \frac{1}{a^n}$**Zero Power:**$a^0 = 1$, provided $a \neq 0$

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**Simplifying Expressions with Positive Indices**

Let's consider simplifying expressions where all the indices are positive integers.

**Example 1:** Simplify $(5x^3)^2$.

**Solution:**

Using the power of a power rule, we have:

$(5x^3)^2 = 5^{(2)} × (x^3)^2$

$= 25x^{(3 × 2)}$

$= 25x^6$

**Example 2:** Simplify $12a^5 ÷ 3a^2$.

**Solution:**

Employing the quotient of powers rule, we get:

$12a^5 ÷ 3a^2 = 12 ÷ 3 × a^{(5 - 2)}$

$= 4a^3$

**Simplifying Expressions with Zero Indices**

Expressions involving zero indices can be simplified using the following rule:

- $a^0 = 1$ (where a ≠ 0)

**Example 3:** Simplify $x^0y^4$.

**Solution:**

Applying the zero index rule, we obtain:

$x^0y^4 = 1 × y^4$

$= y^4$

**Simplifying Expressions with Negative Indices**

Expressions with negative indices can be simplified using the following rule:

- $a^{-n} = \frac{1}{a^n}$ (where a ≠ 0)

**Example 4:** Simplify $2x^{-3}y^2$.

**Solution:**

Using the negative index rule, we have:

$2x^{-3}y^2 = 2 × (\frac{1}{x^3}) × y^2$

$= \frac{2y^2}{x^3}$

**Combining Like Terms with Indices**

When simplifying expressions, it's essential to remember that we can only combine terms with **identical bases and indices**.

**Example 5:** Simplify $3x^2 + 5x^2 - 2x^2$.

**Solution:**

Since all the terms have the same base (x) and index (2), we can simply combine them:

$3x^2 + 5x^2 - 2x^2 = (3 + 5 - 2)x^2$

$= 6x^2$

**Important Points to Remember**

**Base and index restrictions:**The laws of indices apply only when the base is non-zero and the denominator in the quotient of powers is non-zero.**Order of operations:**Remember to follow the order of operations (PEMDAS) when simplifying expressions involving multiple mathematical operations.

**Practice Questions**

**Question 1:** Simplify the expression: $(2a^2b)^3$.

**Solution:**

Following the power of a power rule:

$(2a^2b)^3 = 2^{3} × (a^2)^3 × b^3$

$= 8a^{(2 × 3)}b^3$

$= 8a^6b^3$

**Question 2:** Simplify the expression: $x^4y^2 ÷ x^2y$.

**Solution:** Using the quotient of powers rule:

$x^4y^2 ÷ x^2y = x^{(4 - 2)} × y^{(2 - 1)}$

$= x^2y$