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Writing Square Roots in Standard Form Calculator

Reviewed by Calculator Editorial Team

This guide explains how to write square roots in standard form, including the process, formulas, and practical examples. Our calculator helps you simplify square roots quickly and accurately.

What is Standard Form?

Standard form for square roots refers to expressing a square root in its simplest radical form. This means:

  • Removing any perfect square factors from the radicand (the number inside the square root)
  • Rationalizing the denominator if the square root is in a fraction
  • Simplifying the expression to its most reduced form

The standard form of a square root is typically written as a product of a perfect square and another square root, or as a single square root with no perfect square factors in the radicand.

How to Write Square Roots in Standard Form

To write a square root in standard form, follow these steps:

  1. Factor the radicand into perfect squares and other factors
  2. Separate the perfect square factors from the remaining factors
  3. Write the square root of the perfect square as a simple integer
  4. Combine the results to form the simplified expression

Formula

For a square root √(a×b), where a is a perfect square, the standard form is:

√(a×b) = √a × √b = n × √b

Where n is the square root of a (√a)

Step-by-Step Example

Let's simplify √(72):

  1. Factor 72: 72 = 36 × 2
  2. 36 is a perfect square (6²)
  3. √(36 × 2) = √36 × √2 = 6√2

The standard form of √72 is 6√2.

Examples

Here are several examples of square roots written in standard form:

Original Expression Standard Form Explanation
√50 5√2 50 = 25 × 2, √25 = 5
√108 6√3 108 = 36 × 3, √36 = 6
√192 8√3 192 = 64 × 3, √64 = 8
√(200/4) 10√(5/4) 200 = 100 × 2, √100 = 10

Note: When dealing with fractions under square roots, you may need to rationalize the denominator by multiplying the numerator and denominator by the conjugate of the denominator.

Common Mistakes

Avoid these common errors when simplifying square roots:

  • Assuming all numbers are perfect squares - only 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 are perfect squares under 100
  • Forgetting to separate the perfect square factor from the remaining radicand
  • Incorrectly calculating the square root of the perfect square factor
  • Not rationalizing denominators when required

Double-check your work by squaring the simplified form to ensure it equals the original radicand.

FAQ

What is the standard form of a square root?
The standard form of a square root is when the expression is simplified to remove any perfect square factors from the radicand and written as a product of an integer and a square root.
How do I know if a number is a perfect square?
A perfect square is a number that can be expressed as the square of an integer. Common perfect squares include 1, 4, 9, 16, 25, 36, 49, 64, 81, and 100.
What if the radicand has no perfect square factors?
If the radicand has no perfect square factors other than 1, then the square root is already in its simplest form.
How do I simplify square roots with variables?
The process is similar to numerical radicands. Factor the expression inside the square root, separate the perfect square factors, and simplify accordingly.
What if I have a fraction under a square root?
You can simplify the numerator and denominator separately, then rationalize the denominator if needed by multiplying numerator and denominator by the conjugate of the denominator.