GCSE Maths Practice: powers-and-roots

Understanding Negative Fractional Indices

Negative fractional indices combine two ideas you already know: fractional powers and negative exponents. Fractional powers link roots and powers through the rule \(a^{m/n} = \sqrt[n]{a^m}\). Negative exponents create reciprocals through \(a^{-k} = \tfrac{1}{a^k}\). When both appear together, they describe the reciprocal of a root or a power, depending on which part of the fraction you focus on.

The Meaning of the Negative Sign

A negative power does not make a number negative. Instead, it inverts the base. For example, \(2^{-3} = \tfrac{1}{2^3} = \tfrac{1}{8}\). The negative exponent simply tells you to take the reciprocal after applying the positive power. This rule remains true even when the exponent is a fraction: \(a^{-m/n} = \tfrac{1}{a^{m/n}}\).

Step-by-Step Method

1. Start by rewriting the negative fractional index as a reciprocal: \(a^{-m/n} = \tfrac{1}{a^{m/n}}\).
2. Convert the fractional power inside the denominator to root form: \(a^{m/n} = (\sqrt[n]{a})^m\).
3. Evaluate the root first if possible, then apply the power.
4. Keep the result as a fraction (reciprocal) instead of a decimal for exact answers.

Worked Examples (Different Numbers)

• \(8^{-2/3} = \tfrac{1}{8^{2/3}} = \tfrac{1}{(\sqrt[3]{8})^2} = \tfrac{1}{4}\)
• \(16^{-3/4} = \tfrac{1}{16^{3/4}} = \tfrac{1}{(\sqrt[4]{16})^3} = \tfrac{1}{8}\)
• \(125^{-2/3} = \tfrac{1}{125^{2/3}} = \tfrac{1}{(\sqrt[3]{125})^2} = \tfrac{1}{25}\)

In every case, the process is the same: turn the negative index into a reciprocal, find the root, then apply the power.

Common Mistakes

• Believing that a negative exponent makes the number negative — it simply inverts it.
• Forgetting to apply both parts of the fractional index: students often take the root but forget to square or cube afterwards.
• Switching numerator and denominator of the fraction by mistake.
• Trying to calculate everything as a decimal, which often introduces rounding errors — always keep exact forms when possible.

Real-Life Applications

Negative fractional indices appear in many scientific formulas. In physics, inverse-square and inverse-cube laws (such as gravity or sound intensity) use negative powers to represent quantities that decrease with distance. In chemistry, reaction rate equations can involve fractional and negative exponents that express how concentration affects rate. In finance, depreciation and discount factors often use powers less than one, while inverse growth is represented with negative exponents. Learning this rule prepares you to interpret exponential models in many fields.

Quick FAQ

• Q1: What does the negative sign mean in a power?
  A1: It means take the reciprocal — place 1 over the positive power of the base.
• Q2: What does the fraction in the power represent?
  A2: The denominator shows which root to take, and the numerator shows which power to raise to.
• Q3: Do negative fractional powers follow the same laws of indices?
  A3: Yes. When multiplying powers with the same base, add exponents — even if they are fractions or negative.

Study Tip

Always deal with the negative sign first by writing the reciprocal, then apply the fractional power rule. Remember: \(a^{-m/n} = \tfrac{1}{(\sqrt[n]{a})^m}\). Practise with both perfect cubes and fourth powers to become confident in recognising when results simplify to exact fractions. Understanding this concept is essential for simplifying algebraic expressions, solving exponential equations, and interpreting inverse relationships in real-world problems.