GCSE Maths Practice: powers-and-roots

Understanding Square Roots of Decimals

Square roots describe the process of finding a number which, when multiplied by itself, produces another number. When the number involves decimals, the principle stays the same — only the scale changes. The result of squaring a decimal will always be smaller than one if the original decimal is less than one.

Concept and Importance

Square roots of decimals are used in many parts of GCSE Maths, especially when dealing with percentages, area, and scale factors. They allow us to move between areas and side lengths, or between growth factors and percentages in proportion problems.

Method to Find a Square Root of a Decimal

1. Write the decimal as a fraction, if it helps — for instance, 0.64 = 64/100.
2. Find the square root of both numerator and denominator separately.
3. Convert back to a decimal if needed.

This method shows that decimals behave exactly like fractions under square roots, as both are simply different ways of expressing parts of a whole.

Worked Examples (Different Numbers)

• \(\sqrt{0.25} = 0.5\) because 0.5 × 0.5 = 0.25.
• \(\sqrt{0.49} = 0.7\).
• \(\sqrt{0.64} = 0.8\).

These examples show a pattern: as decimals increase toward one, their square roots increase as well.

Common Mistakes

• Assuming the answer must be greater than one — it is smaller when the original number is less than one.
• Forgetting to square both digits when checking a result.
• Misplacing the decimal point when converting between fractions and decimals.

Real-Life Applications

Square roots of decimals are widely used in real-world contexts. For example, when calculating the side length of a small square tile whose area is a fraction of one square metre, or when scaling down models in design and architecture. They also appear in financial calculations, such as compound growth where small rate changes accumulate over time.

Quick FAQ

• Q1: Why do square roots of decimals matter in GCSE Maths?
  A1: They show how proportional reasoning and area relationships work even with numbers smaller than one.
• Q2: Do all decimals have square roots?
  A2: Yes — every positive decimal has one positive square root, though some are irrational (non-terminating).
• Q3: How can I check if my answer is correct?
  A3: Multiply your result by itself — if you return to the original decimal, it’s correct.

Study Tip

Practise squaring decimals like 0.5, 0.6, 0.7, 0.8, and 0.9 to see how their squares behave. You’ll notice that each result gets smaller, which helps build intuition for square roots and area relationships. Understanding these patterns is vital in GCSE topics involving indices, geometry, and percentages.