GCSE Maths Practice: powers-and-roots

Question 9 of 10

This question tests your understanding of the zero power rule — a fundamental concept in the Powers and Roots topic. Recognising this rule helps simplify many GCSE index problems.

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\( \begin{array}{l} \text{What is } 5^0? \end{array} \)

Choose one option:

When you see a zero power, recall that any non-zero base equals one. This rule keeps index calculations consistent.

Understanding the Zero Power Rule

The zero power rule is one of the most useful shortcuts in mathematics. It tells us that any non-zero number raised to the power of zero equals one. Although this rule may look like magic at first, it comes directly from the laws of indices.

Why Does This Work?

Think about the division rule for powers: when dividing numbers with the same base, you subtract their exponents. For example, \(a^5 \div a^5 = a^{5-5} = a^0\). But any number divided by itself equals one, so \(a^0 = 1\). This reasoning works for every non-zero value of a.

Step-by-Step Method

  1. Write down the rule: \(a^m \div a^m = a^{m-m} = a^0\).
  2. Recognise that dividing a number by itself gives one.
  3. Therefore, the expression equals one for all non-zero numbers.

This reasoning ensures the laws of indices stay consistent for all powers, including zero and negatives.

Worked Examples (Different Bases)

  • \(2^0 = 1\)
  • \(10^0 = 1\)
  • \((-7)^0 = 1\)
  • \(\left(\tfrac{1}{3}\right)^0 = 1\)

Notice that the value is always one, regardless of whether the base is positive, negative, or fractional — as long as it is not zero.

Common Misconceptions

  • Some students think any power of zero equals zero — that’s incorrect. Only 0 to a positive power equals zero.
  • Confusing \(0^0\), which is undefined, with the zero power rule for other numbers.
  • Applying the rule without understanding its logical base from the laws of indices.

Real-Life Connections

The zero power rule appears in many GCSE contexts, including scientific notation, algebraic simplification, and exponential functions. For example, when graphing exponential decay, the y-intercept often corresponds to the point where the exponent is zero, giving a value of one.

Quick FAQ

  • Q1: Does this rule work for negative bases?
    A1: Yes, as long as the base is not zero, the result is still one.
  • Q2: What about 00?
    A2: 00 is undefined — it doesn’t follow the same rule because division by zero is impossible.
  • Q3: How does this help in algebra?
    A3: It simplifies terms such as \(x^0\) to 1, making expressions easier to handle.

Study Tip

To remember the rule, think of any number divided by itself: it always equals one. The zero exponent is just a shorthand for that concept. Practise simplifying powers with zero exponents to gain confidence in algebraic manipulation and exponential equations.