Grade 7 Mathematics NUMBERS – Squares and Square roots Notes

A square of a number is the number multiplied by itself. We write the square of n as n². Example: 4² = 4 × 4 = 16.

In geometry, if a square has side length s, its area = s². This matches the number idea.

The square root of a number x is a number y such that y² = x. We write square root using √. So √25 = 5 because 5² = 25.

A number may have two square roots: one positive and one negative (e.g. ±5 for 25), but usually we use the positive root (principal root) for positive numbers.

• Check the list of common perfect squares (above).
• If a number is the square of an integer, it is a perfect square (e.g. 49 = 7²).
• If not, it is not a perfect square (e.g. 20 is not a perfect square because no whole number squared gives 20).

To find √18: find two perfect squares around 18. 16 (4²) and 25 (5²). So √18 is between 4 and 5. Since 18 is closer to 16 than 25, √18 is a little above 4 (≈ 4.24).

Quick rule: If x is between a² and (a+1)², then √x is between a and a+1.

• Use a calculator for exact decimals.
• Use prime factorisation for exact whole roots: e.g. √144 = √(2⁴ × 3²) = 2² × 3 = 12.
• Use approximation between perfect squares for quick estimates.

1. Find 7². Answer: 7 × 7 = 49.
2. Find √36. Answer: 6, because 6 × 6 = 36.
3. Estimate √50. 7²=49 and 8²=64, so √50 is slightly above 7 (≈7.07).
4. Exact by factor method: √225 = √(15 × 15) = 15.

1. What is 9²?
2. What is √81?
3. Is 45 a perfect square? Explain.
4. Estimate √20 (between which integers? approximate value).
5. Use prime factors to find √196.