Square Root Calculator is a free online tool provided by 365 Calcs.

Please enable JavaScript in your browser to complete this form.
We'll send the results to your email for easy reference.

How To Use The Square Root Calculator

  1. Enter a Number:
    • In the ‘Enter a Number’ field, type the number for which you want to calculate the square root.
  2. Square Root:
    • After entering the number, the square root (if it’s a perfect square) or the nearest square root approximation will be displayed in the ‘Square Root’ field.
  3. Verdict:
    • The ‘Verdict’ field will inform you whether the entered number is a perfect square or not.
  4. Enter Your Email:
    • If you would like to have the results sent to you, enter your email address in the ‘Email’ field.
  5. Send Results:
    • Click the ‘Send Results’ button to submit the information. If you’ve entered an email, the results will be sent to that address for your reference.

Frequently Asked Questions

What is the rule for calculating square roots?

Calculating the square root of a number is the process of finding a value that, when multiplied by itself, gives you the original number. The square root of a number ( x ) is often denoted as ( \sqrt{x} ). Here are the basic rules and methods for calculating square roots:

  1. Perfect Squares: For some numbers, the square root is an integer. These numbers are called perfect squares. For example, ( \sqrt{25} = 5 ) because ( 5 \times 5 = 25 ).
  2. Estimation: For numbers that are not perfect squares, you can estimate the square root. You find the nearest perfect squares that the number falls between and estimate the root based on its position between them.
  3. Long Division Method: This is a manual method similar to traditional division that allows you to find the square root of any number to as many decimal places as needed. It’s a bit complex to explain in a brief format but works well for exact calculations.
  4. Prime Factorization: You can factor the number into its prime factors and pair the primes. The square root is then the product of one number from each pair. For example, to find ( sqrt{72} ):
    • Factor 72 into prime factors: (72 = 2*2*2*3*3).
    • Pair the factors: ( (2*2) * (2) * (3 * 3) ).
    • Take one from each pair: ( 2 * 3 ).
    • Multiply the unpaired factors: ( 2 * 3 * sqrt{2} ).
    • The square root is ( 6 sqrt{2} ), since ( 6 * 6 = 36 ) and you still have an unpaired ( 2 ).
  5. Using a Calculator: Most calculators have a square root function. You just enter the number and press the square root button (( sqrt{} )).
  6. Approximation Algorithms: There are various algorithms like Newton’s method (also known as the Newton-Raphson method) that can be used to approximate square roots, which is how many digital calculators perform the operation internally.
  7. Digital Calculators and Software: In practice today, square roots are most often calculated using digital calculators or software, which can quickly and accurately find square roots to many decimal places.

These are some of the methods and rules you can use to calculate square roots, depending on the level of precision required and whether you are doing it manually or using a tool.

Can you work out square root without a calculator?

Yes, you can calculate the square root of a number without a calculator using several manual methods. Here are a couple of the most common ones:

Estimation Method:

  1. Identify Perfect Squares: Find the two perfect square numbers it falls between. For example, if you’re finding the square root of 50, it falls between the perfect squares of 49 (7^2) and 64 (8^2).
  2. Estimate: Since 50 is closer to 49, you can estimate that the square root of 50 is slightly more than 7.

Long Division Method (for square roots):

  1. Set up a division: Write the number in pairs of digits, starting from the decimal point and going both left and right. For whole numbers, start from the units place.
  2. Find the largest square: Find the largest square smaller than or equal to the first pair. This is the first digit of the root.
  3. Subtract and bring down the next pair: Subtract the square from the first pair, bring down the next pair of digits, and double the root’s first digit for the divisor’s left part.
  4. Find the next digit: Find a digit which, when added to the divisor and multiplied by the same digit, is less than or equal to the dividend. Subtract the result from the dividend and bring down the next pair of digits.
  5. Repeat: Continue the process until you have achieved the desired level of accuracy.

Prime Factorization (for integers):

  1. Factor the number: Break down the number into its prime factors.
  2. Form pairs of prime factors: Group the prime factors into pairs of the same number.
  3. Take one from each pair: For every pair of the same number, take one number out of the square root symbol.
  4. Multiply for the root: Multiply the numbers you took out for the square root. If there are any prime numbers left inside the square root that could not be paired, they stay inside the square root.

For example, if you’re finding the square root of 72:

  • Prime factorize: ( 72 = 2 \times 2 \times 2 \times 3 \times 3 )
  • Pair the factors: ( (2 \times 2) \times (3 \times 3) \times 2 )
  • Take one from each pair: ( 2 \times 3 \times \sqrt{2} )
  • The square root is ( 6\sqrt{2} ).

Graphical Method:

For a more visual approach, you can also estimate square roots using a graph by plotting the function ( f(x) = x^2 ) and finding the point where ( f(x) = ) your number.

These methods can be time-consuming and require practice to perform efficiently, but they can be quite effective for learning purposes or when a calculator is not available.

Does The Power Rule Apply To Square Roots?

Yes, the power rule does apply to square roots. When you take the square root of a number that’s raised to a power, you can simplify it by dividing the exponent by two. Conversely, if you raise a square root to a power, you can apply the power to the number inside the square root. These rules help you work with square roots in algebraic expressions and come in handy especially when you need to simplify complex expressions or solve equations.

Is 64 a square number?

Yes, 64 is a square number. It is the square of 8, since 8 multiplied by 8 equals 64. Square numbers are the product of an integer multiplied by itself, and 64 fits this definition perfectly.