Radical Simplifier Calculator
Simplify square roots instantly to their exact simplest radical form.
Quick Examples
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About Radical Simplifier Calculator
What is a Radical?
A radical represents the root of a number. Customarily, when we talk about simplifying radicals, we are dealing with square roots (represented by the symbol $\sqrt{x}$).
How to Simplify Radicals
Simplifying a radical means extracting any perfect square factors from the radicand (the number under the radical symbol) and moving them outside the radical.
The Process:
- Find Factors: Break the radicand down into its prime factors or look for perfect square factors (like 4, 9, 16, 25, 36, etc.).
- Extract Perfect Squares: If a perfect square is a factor, take its square root and place that number outside the radical symbol.
- Leave the Rest: The remaining factors stay inside the radical.
Example: Simplifying $\sqrt{72}$
- The factors of 72 include $36 \times 2$.
- 36 is a perfect square (since $6^2 = 36$).
- We can rewrite the expression as $\sqrt{36 \times 2}$.
- Taking the square root of 36 gives 6, which moves outside the radical: $6\sqrt{2}$.
Understanding the Results
This calculator will output three different types of results depending on your input:
- Already Simplified: If the number has no perfect square factors (e.g., $\sqrt{7}$), the radical is already in its simplest form.
- Perfect Square: If the number is a perfect square (e.g., $\sqrt{144}$), the result is an exact integer ($12$).
- Simplified Radical: If the number has a perfect square factor but isn't a perfect square itself (e.g., $\sqrt{72}$), it will output the mixed format like $6\sqrt{2}$.
FAQ
Can a radical be negative? The principal (customary) square root of a positive number is always positive. Taking the square root of a negative numbers results in imaginary/complex numbers, which are typically denoted with an $i$.
Why simplify radicals instead of using decimals? Simplified radicals provide an exact answer. Decimals are usually rounded approximations, which can lead to inaccuracies in complex multi-step physics or engineering problems.
Disclaimer: This calculator is provided for educational and illustrative purposes. While every effort is made to ensure accuracy, complex mathematical results should be verified for critical applications.