EDEXCEL GCSE HIGHER MATHS - What are Indices (Fractional and Negative)

Edexcel GCSE Higher Maths - Fractional and Negative Indices

What are Indices?

Indices are a way of representing repeated multiplication. For example, 2³ means 2 multiplied by itself three times (2 x 2 x 2 = 8).

Fractional Indices

A fractional index represents a root. For example:

• x½ = ?x (square root of x)
• x? = ³?x (cube root of x)
• x¼ = ??x (fourth root of x)

Rules for Fractional Indices

1. xm/n = (n?x)m This means you find the nth root of x, and then raise it to the power of m.

2. xm/n = n?xm This means you raise x to the power of m, and then find the nth root of the result.

Examples

• 82/3 = (³?8)2 = 22 = 4
• 163/4 = ??163 = 23 = 8

Negative Indices

A negative index represents the reciprocal of the corresponding positive index. For example:

• x-1 = 1/x
• x-2 = 1/x²

Rules for Negative Indices

1. x-n = 1/xn

2. 1/x-n = xn

Examples

• 2-3 = 1/2³ = 1/8
• 5-2 = 1/5² = 1/25

Simplifying Expressions with Fractional and Negative Indices

Use the rules above to simplify expressions with fractional and negative indices. Remember to follow the order of operations (BODMAS).

Examples

• (x1/2)4 = x2
• x3 * x-2 = x1 = x
• (2x2)-1 = 1/(2x2) = 1/2 * x-2 = 1/2x2

Practice Problems

1. Simplify 93/2
2. Evaluate 16-1/2
3. Simplify (2x-1)3

Answers

1. 27
2. 1/4
3. 8/x3

Key Points to Remember

• Fractional indices represent roots.
• Negative indices represent reciprocals.
• Use the rules to simplify expressions with fractional and negative indices.