How can I estimate √28 without a calculator?

Estimating √28 without a calculator is an excellent mental math exercise that develops number sense and mathematical intuition. The most straightforward method uses your knowledge of perfect squares to bracket the value between two known square roots. Since you know that 5² = 25 and 6² = 36, you can immediately determine that √28 must fall between 5 and 6. Furthermore, since 28 is much closer to 25 (only 3 units away) than to 36 (8 units away), you can estimate that √28 is closer to 5 than to 6, giving you a rough estimate of approximately 5.3.

For a more refined estimate, you can use linear interpolation, a technique that assumes the square root function increases at a roughly constant rate between two consecutive perfect squares (though this is an approximation, as the square root function is actually curved). The calculation works like this: 28 is 3 units above 25, and the distance from 25 to 36 is 11 units. Therefore, 28 is located 3/11 (approximately 0.27) of the way from 25 to 36. If we assume the square root function increases proportionally, √28 should be approximately 0.27 of the way from 5 to 6. Calculating 5 + (0.27 × 1) = 5.27, which is very close to the true value of 5.29. This method provides a remarkably accurate estimate considering it requires only mental arithmetic and knowledge of basic perfect squares.

Another powerful estimation technique uses the formula: if you want to find √N and you know a nearby perfect square a² ≈ N, then √N ≈ a + (N - a²)/(2a). For √28, we can use a = 5 (since 5² = 25 is close to 28). Applying the formula: √28 ≈ 5 + (28 - 25)/(2 × 5) = 5 + 3/10 = 5.3. This gives us a quick and reasonably accurate estimate. The formula works because it's based on the calculus concept of linear approximation, but you don't need to understand calculus to use it effectively. With practice, this calculation can be performed mentally in just a few seconds.