Negative and Fractional Indices
Grade 6Negative indices — the reciprocal connection
A negative index means "take the reciprocal". That is it. a⁻ⁿ = 1 ÷ aⁿ. So 2⁻³ = 1 ÷ 8 = ⅛.
The reason this works follows from the division law: a² ÷ a⁵ = a^(2−5) = a⁻³. But written out long, a² ÷ a⁵ = 1 ÷ a³. So a⁻³ must equal 1/a³.
Fractional indices — roots and powers combined
A fractional index connects powers to roots:
a^(1/n) = the nth root of a
a^(m/n) = (nth root of a)^m
For a^(m/n): take the root first, then raise to the power. Roots make numbers smaller, so do that step first — it keeps the numbers manageable.
Worked example
EXAMPLE
Without a calculator, evaluate: (a) 8^(1/3) (b) 27^(2/3) (c) 4⁻²
(a) 8^(1/3) = cube root of 8 = 2.
(b) 27^(2/3): cube root of 27 = 3, then 3² = 9.
(c) 4⁻² = 1 ÷ 4² = 1 ÷ 16 = 1/16.
The zero index
Any base (except zero) raised to the power of 0 equals 1. So 7⁰ = 1, x⁰ = 1, and even (−5)⁰ = 1. This follows from the division law: a^n ÷ a^n = a⁰ = 1.
Try this one
Simplify the following expressions.
(a) \( x^5 \times x^3 \) [1]
(b) \( \dfrac{y^{10}}{y^4} \) [1]
(c) \( (z^3)^4 \) [1]
Nova's hint:
Try to Solve it with Nova?
These questions use the index laws (also called laws of exponents).
Ask yourself: 'Which index law applies — am I multiplying, dividing, or raising a power to a power?'
For example, \( a^m \times a^n = a^{m+n} \), and \( (a^m)^n = a^{mn} \).
