# Simplest Radical Form Calculation

## What is simplified radical form in math?

Simplified radical form is when a number under the radical is indivisible by a perfect square other than 1.

In mathematics, expressing numbers in **simplest radical form** involves representing square roots or other radicals
in a way that eliminates any unnecessary factors or radicals. This process simplifies expressions and makes them
more manageable and readable.

## Examples of Simplest Radical Form

There are four criteria for a radical to be in Simplest Form.

- The radicand contains only powers less than the index.
- The index is as small as possible.
- The denominator has been rationalized.
- No fractions occur under the radical.

Let's explore a few examples to better understand how to calculate and express numbers in their simplest radical form.

Expression | Simplest Radical Form |
---|---|

√18 | 3√2 |

√50 | 5√2 |

√72 | 6√2 |

## Graphical Representation

Below are graphical representations of the examples above, illustrating the simplification process.

The graphical representation visually emphasizes the simplification, where each bar represents the original expression, and the colored portion signifies the simplified form.

Calculating simplest radical form is a crucial skill in mathematics. It allows for more straightforward manipulation of expressions and clearer communication of mathematical concepts.

## Simplifying Radical Expressions with Square Root

For simplifying radical expressions with square root, let us consider an example. Consider the radical expression √486. We will simplify this radical expression into the simplest form until no further simplification can be done.

**Step 1:** Find the factors of the number under the radical.

486 = 3 × 3 × 3 × 3 × 3 × 2

**Step 2:** Write the number under the radical as a product of its factors as powers of 2.

486 = 32 × 32 × 3 × 2

**Step 3:** Write the factors outside the radical which have the power 2.

√486 = √(32 × 32 × 3 × 2) = 3 × 3 √(3 × 2) = 9√(3 × 2)

**Step 4:** Simplify the radical until no further simplification can be done

√486 = 9√(3 × 2) = 9√6. No further multiplication can be done now.

Hence, we have simplified the radical expression √486 as 9√6 and it cannot be simplified more.

## Calculation Table:

Square Root | Answer | Calculation |
---|---|---|

√58 | Square root of 58 is the simplest radical form. | √2 x √29 |

√56 | 2√14 | √2 x √2 x √2 x √7 |

√11 | Square root of 11 is the simplest radical form. | √11 |

√165 | Square root of 165 is the simplest radical form. | √3 x √5 x √11 |

√694 | Square root of 694 is the simplest radical form. | √2 x √347 |

√624 | 4√39 | √2 x √2 x √2 x √2 x √3 x √13 |

√9568 | 4√598 | √2 x √2 x √2 x √2 x √2 x √13 x √23 |

√5819 | 23√11 | √11 x √23 x √23 |

√5338 | Square root of 5338 is the simplest radical form. | √2 x √17 x √157 |