Calculator
Example Data Table
| Input | Largest Square Factor | Extracted Form | Decimal Value |
|---|---|---|---|
| √72 | 36 | 6√2 | 8.485281 |
| √121 | 121 | 11 | 11 |
| 3√80 | 16 | 12√5 | 26.832816 |
| √-50 | 25 | 5√2i | 7.071068i |
| √0.25 | Approximation | Decimal root | 0.5 |
Formula Used
The calculator uses the largest perfect-square factor of the radicand.
√n = √(a² × b) = a√b
Here, a² is the largest perfect-square factor of n. The value b is the square-free remainder.
When an outside coefficient is used, the formula becomes:
c√n = c × a√b
For negative radicands in complex mode, the calculator applies:
√(-n) = i√n
Example: √72 = √(36 × 2) = 6√2.
How to Use This Calculator
- Enter the radicand. This is the number under the square root sign.
- Enter an outside coefficient if your expression already has one.
- Select decimal places for the approximate answer.
- Choose how negative radicands should be handled.
- Select a rounding method for decimal output.
- Add an optional comparison radicand if needed.
- Press Calculate to view the result above the form.
- Use the CSV or PDF button to save your work.
Understanding Square Root Extraction
Why extraction matters
Square root extraction makes radicals easier to read. It also makes answers cleaner. Many math problems need exact form. Decimal form is helpful, but it can hide structure. Exact form keeps the radical part visible. This helps in algebra, geometry, and trigonometry.
Perfect-square factors
The key idea is simple. A square root can be split when the radicand has a perfect-square factor. For example, 72 contains 36. Since the square root of 36 is 6, the expression becomes 6√2. The 2 stays inside because it has no square factor greater than 1.
Using coefficients
Some expressions already include a coefficient. For example, 3√80 has 3 outside the radical. The calculator first extracts √80 as 4√5. Then it multiplies 3 by 4. The final exact form is 12√5. This keeps the expression compact and correct.
Decimal support
Decimals are useful for measurement and checking work. The calculator gives a rounded decimal result. You can choose precision from zero to twelve places. You can also choose nearest, floor, ceiling, or truncate. This is helpful when a class or project requires a certain rounding rule.
Complex roots
Negative radicands do not have real square roots. In complex mode, the calculator uses the imaginary unit. The rule is √(-n) = i√n. So √-50 becomes 5√2i. This option helps students move between real and complex number systems.
Better checking
The step list shows how the answer was created. It lists the largest square factor, the extracted outside factor, and the remaining radicand. These details make it easier to spot mistakes. They also make the result useful for homework review and classroom notes.
FAQs
1. What does extract square root mean?
It means removing perfect-square factors from inside a radical. For example, √72 becomes 6√2 because 36 is the largest perfect-square factor of 72.
2. What is the radicand?
The radicand is the number inside the square root symbol. In √45, the radicand is 45. The calculator simplifies that number when possible.
3. Why does the calculator show exact and decimal results?
Exact form is best for algebraic work. Decimal form is useful for measurement, estimation, and checking. Showing both gives a clearer answer.
4. Can this calculator handle negative radicands?
Yes. Choose complex mode to use imaginary numbers. If real-only mode is selected, negative radicands are rejected because they have no real square root.
5. What is a square-free remainder?
It is the number left inside the radical after all possible square factors are extracted. In 6√2, the square-free remainder is 2.
6. How does the outside coefficient work?
The outside coefficient multiplies the extracted square root factor. For 3√80, √80 becomes 4√5, then 3 × 4 gives 12√5.
7. Why is exact simplification limited for some decimals?
Decimal radicands may not simplify neatly as integer radicals. The calculator still gives a decimal result, but exact extraction is focused on whole numbers.
8. Can I save the result?
Yes. After calculating, use the CSV button for spreadsheet data. Use the PDF button for a printable summary of the result and steps.