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How to Square Root A Number on Calculator

Reviewed by Calculator Editorial Team

Calculating square roots is a fundamental mathematical operation with applications in geometry, algebra, and many other fields. This guide explains how to find square roots using both calculators and manual methods, along with practical examples and common uses.

How to Use a Calculator for Square Roots

Most scientific and graphing calculators have a dedicated square root function. Here's how to use it:

Formula: √x = y where y² = x

  1. Turn on your calculator and clear any previous calculations.
  2. Enter the number you want to find the square root of.
  3. Press the square root button (often labeled √ or √x).
  4. Press the equals (=) button to display the result.

For example, to find the square root of 25:

Example: √25 = 5 because 5 × 5 = 25

If your calculator doesn't have a dedicated square root button, you can use the exponent function (often labeled yˣ or ^) by entering 0.5 as the exponent. For example, to find √25, you would enter 25^(0.5).

Calculator Tips

  • Always check that your calculator is in the correct mode (usually "DEG" for degrees or "RAD" for radians, though this doesn't affect square roots).
  • For very large or very small numbers, scientific notation may be used.
  • If you're working with negative numbers, remember that square roots of negative numbers are not real numbers (they are complex numbers).

Manual Methods for Square Roots

While calculators are convenient, understanding manual methods can help you verify results and understand the concept better.

Prime Factorization Method

This method works well for perfect squares:

  1. Factor the number into its prime factors.
  2. Pair the prime factors and take one from each pair.
  3. Multiply the numbers from each pair to get the square root.

Example: Find √36

36 = 2 × 2 × 3 × 3

√36 = √(2 × 2 × 3 × 3) = 2 × 3 = 6

Long Division Method

This method works for any positive real number:

  1. Group the digits in pairs from the decimal point, starting from the right.
  2. Find the largest number whose square is less than or equal to the first group.
  3. Subtract its square from the group and bring down the next pair.
  4. Double the current result and find a digit to append that forms a new number whose square is less than the new dividend.
  5. Repeat until you have the desired precision.

Example: Find √2 to 3 decimal places

1. 1² = 1 ≤ 2, so first digit is 1. Remainder: 2-1=1

2. Bring down 00 → 100. Double 1 → 2. Find largest digit d where (20 + d)² ≤ 100: d=4 (24²=576)

3. Remainder: 100-576=424. Bring down 00 → 42400

4. Double 14 → 28. Find largest digit d where (280 + d)² ≤ 42400: d=0 (280²=78400 too large)

Result: √2 ≈ 1.414

Common Applications of Square Roots

Square roots have many practical applications in various fields:

Geometry

  • Finding the length of a side of a square when the area is known.
  • Calculating distances between points in coordinate geometry.
  • Determining the radius of a circle when the area is known.

Algebra

  • Solving quadratic equations.
  • Simplifying expressions with square roots.
  • Working with complex numbers.

Everyday Life

  • Calculating the diagonal of a rectangle.
  • Determining the optimal dimensions for a garden or room.
  • Understanding growth rates in financial calculations.

Example Problem: A square garden has an area of 64 square meters. What is the length of one side?

Solution: √64 = 8 meters

Frequently Asked Questions

What is the difference between a square root and a square?
The square of a number is that number multiplied by itself (e.g., 5² = 25). The square root of a number is a value that, when multiplied by itself, gives the original number (e.g., √25 = 5).
Can I find the square root of a negative number?
In real numbers, no. The square root of a negative number is not a real number. However, in complex numbers, the square root of a negative number is an imaginary number (e.g., √-1 = i).
What if my calculator doesn't have a square root button?
You can use the exponent function with 0.5 as the exponent. For example, to find √25, enter 25^(0.5).
How do I know if a number is a perfect square?
A number is a perfect square if it can be expressed as the square of an integer. For example, 36 is a perfect square (6²), but 37 is not.
What are some real-world uses of square roots?
Square roots are used in geometry to find lengths, in algebra to solve equations, in physics to calculate distances and forces, and in finance for risk assessment.