Mentally Get the Square Root of Any Number Quickly

Whether estimating perimeters of a right triangle, finding the roots of a quadratic equation, or taking square roots of big numbers just for fun, it is useful to be able to find the decimal equivalent or reduce a radical (square root).

Steps

  1. Know some easy square numbers, such as the first 15. You can multiply the number by 100 and the square root will be equal to the original number's square root times 10. That is, 64^(1/2) = 8, and 6400^(1/2) = 80. Expressed as variables, it is x^(1/2) = y <==> (x*100)^(1/2) = y*10
  2. If you want to know the decimal equivalent of the number, know the square roots of a few simple numbers to 3-4 points of precision.
  3. For some low numbers, you can estimate the square root by looking for square numbers around it. For example, if you want to know the square root of 8, know that 4 and 9 are square numbers, and their square roots are 2 and 3 respectively. 8 is closer to 9 than 4, so the square root is probably closer to 3. Since 4 and 9 are so close to each other, the square root of 8 is probably around 2.8 (the actual square root is 2.828). Having a good number sense will help you estimate the decimal equivalent more accurately.
  4. Know how to reduce radicals. You can take the square root of any number by taking the square root of some of its factors. If x = zy, then the square root of x = z^(1/2)*y^(1/2). Let's use 8 as an example again. 8 = 4*2, so the square root of 8 is equal to the square root of 4 times the square root of 2. Since we know the square root of 4 is 2 and the square root of 2 is about 1.414, you can multiply these to get 2.828, which is the square root of 8. Once you get down to smaller, more manageable numbers, you can more accurately find the square root.



Tips

  • The first 15 square numbers (in order) are: 1, 4, 9,16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
  • Develop a good number sense by paying attention in math class.
  • The square roots of the first 3 non-square numbers are (note that these are irrational numbers, so don't think they end after 3 decimal points!): 2^(1/2) = 1.414; 3^(1/2) = 1.732; 5^(1/2) = 2.23606.
  • Using a calculator is more efficient, but knowing how to simplify the radical is useful for a pre-calculus or trigonometry class.

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