Solve Square Root Problems

To solve square root problems, especially if we’re looking for a simple approach, here are some detailed methods we can consider:

1. Understanding Square Roots

• The square root of a number ( x ) is a value ( y ) such that ( y² = x ).
• For example, the square root of 16 is 4, because ( 4² = 16 ).

2. Using Estimation

• Identify Perfect Squares: Recognize the nearest perfect squares around the number. For example, for ✓20} ):
  • The nearest perfect squares are ( 16 ) (which is 4²) and ( 25 ) (which is (5²).
  • Since ( 20 ) is between ( 16 ) and ( 25 ), we can estimate that ✓20 is between ( 4 ) and ( 5 ).

3. Long Division Method

• This is a manual method for finding square roots:
  1. Pair the digits of the number starting from the decimal point (for whole numbers, pair from the right).
  2. Find the largest square less than or equal to the first pair (or single digit).
  3. Subtract the square from the pair and bring down the next pair.
  4. Double the number found so far, and find a digit ( d ) such that ( (2 x current number} + d) x d ) is less than or equal to the new number we have.
  5. Repeat until we reach the desired precision.

4. Using a Calculator

• For quick results, most calculators have a square root function. Simply input the number and press the square root key.

5. Using Exponents

• We can express square roots in terms of exponents:
  ✓x = x^1/2
• This can be useful in programming or when using certain calculators.

6. Special Cases

• Remember some square roots are easy to compute:
  • ✓0 = 0
  • ✓1 = 1
  • ✓4 = 2
  • ✓9 = 3
  • ✓16 = 4 , etc.

Conclusion

The easiest method often depends on the context and the tools we have available. For quick calculations, a calculator is the simplest method, while estimation can be useful for mental math. Understanding the underlying principles will make the process easier and more intuitive.

Solve a Square root Problem

1.  How find the Square root of 256.

                     Prime factorisation of 256 is

256 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2

By pairing the prime factors we get,

256 = 2 x 2 x 2 x 2 x 2 x 2  2 x 2 

or        256 = (2 x 2 x 2 x 2)²

 Therefore,  √256 = 2 x 2 x 2 x 2  = 16  solution.

2. How find the Square root of 6400.

               Prime factorisation of 6400 is

          6400 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5

By pairing the prime factors we get,

6400 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5

or     256 = (2 x 2 x 2 x 2 x 5)²

Therefore,  
              √6400 = 2 x 2 x 2 x 2 x 5 = 80  solution.