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Understanding perfect squares is crucial when it comes to estimating square roots, especially without the aid of a calculator. A perfect square is an integer that can be expressed as the product of an integer with itself. For example, 25 is a perfect square because it can be written as 5 \times 5. In general, if you multiply an integer n by itself to get n \times n, the result is a perfect square, which we denote as n^2.

Recognizing perfect squares is handy because they serve as boundary markers when estimating the square roots of numbers that are not perfect squares. For instance, in our exercise, we considered perfect squares close to 70, which were 64 (8^2) and 81 (9^2), to frame our estimation for \( \sqrt{70} \).

When it comes to square root approximation, non-calculator math skills come into play. After identifying the perfect squares nearest to the target number, the next step is to interpolate between them. This involves a bit of educated guessing, as well as an understanding of how numbers within this range behave.

To refine our estimate, we consider not just the integers but also the decimal places. The square root of 70 must lie somewhere between 8 and 9 since these are the whole number square roots of the perfect squares closest to 70. We might estimate that because 70 is closer to 64 than to 81, the square root should be closer to 8 than to 9. We can improve our estimate further by considering the difference between 70 and these perfect squares. The closer the number is to the smaller perfect square, the smaller the extra decimal should be. In our example, a reasonable estimate placed the value at \(8.5\), halfway between 8^2 and 9^2 as a starting point.

Being able to tackle math problems without a calculator is a valuable skill, enhancing number sense and mental calculation abilities. When estimating square roots, there are a few non-calculator techniques that can be useful. The first is approximation, as we used in our example. Another is to understand the concept of linear interpolation, which helps us approximate values that lie between two known values.

Even when a calculator is available, understanding the underlying concepts can give you a better sense of numbers and improve your ability to estimate and check work for reasonableness. Non-calculator math skills promote a more profound understanding of how numbers work together and develop critical thinking skills that are beneficial beyond mathematics.