GCSE Maths Practice: powers-and-roots

Question 8 of 10

This question tests your understanding of negative powers — an important part of the Powers and Roots topic in GCSE Maths. Negative indices represent reciprocals, not negative numbers.

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\( \begin{array}{l} \text{What is } 2^{-3}? \end{array} \)

Choose one option:

When you see a negative power, flip the base to the denominator. This turns the power positive and simplifies the calculation.

Understanding Negative Indices

Negative powers do not make numbers negative. Instead, they represent reciprocals. The rule is \(a^{-n} = \tfrac{1}{a^n}\), where a is any non-zero number. This rule ensures the laws of indices remain consistent for all powers, including negative and zero exponents.

Why This Rule Works

The relationship between powers can be seen from patterns. For example, \(2^3 = 8\), \(2^2 = 4\), \(2^1 = 2\), \(2^0 = 1\). Each time the power decreases by one, the result is divided by 2. Continuing that pattern gives \(2^{-1} = \tfrac{1}{2}\), \(2^{-2} = \tfrac{1}{4}\), and \(2^{-3} = \tfrac{1}{8}\). The values get smaller as the powers become negative.

Step-by-Step Method

  1. Write the base number and note the negative power.
  2. Use the reciprocal rule: \(a^{-n} = \tfrac{1}{a^n}\).
  3. Find the positive power \(a^n\).
  4. Place it as the denominator of a fraction with 1 on top.

Worked Examples (Different Numbers)

  • \(3^{-2} = \tfrac{1}{9}\)
  • \(10^{-1} = \tfrac{1}{10}\)
  • \(5^{-4} = \tfrac{1}{625}\)

Notice that the result is always a fraction less than one for positive bases.

Common Mistakes

  • Thinking that a negative power makes the result negative. It does not — it makes it a fraction.
  • Forgetting to apply the reciprocal step.
  • Mixing up \(a^{-n}\) with \(-a^n\). The negative sign in the power affects the position, not the sign of the number.

Real-Life Applications

Negative powers appear in scientific notation and standard form. For example, \(10^{-3}\) represents 0.001. This helps express very small quantities, such as measurements in physics, chemistry, and finance, where reciprocals of large numbers are needed.

Quick FAQ

  • Q1: Why does a negative power mean reciprocal?
    A1: Because each step down in exponent divides by the base instead of multiplying.
  • Q2: Can negative powers be used with decimals or fractions?
    A2: Yes, the rule applies to any non-zero base.
  • Q3: What happens with \(a^{-0}\)?
    A3: There is no such thing — the power 0 always gives 1, not a reciprocal.

Study Tip

Practise writing sequences of powers such as \(2^3, 2^2, 2^1, 2^0, 2^{-1}, 2^{-2}\). Spotting the pattern helps remember that each step down divides by the base. Understanding this connection makes exponent rules much easier.