Calculation Result
Equivalent Forms
| Form | Value |
|---|---|
| Original expression | √(625/98) |
| Reduced radicand fraction | √(625/98) |
| Separated square roots | √625/√98 |
| Extracted radical form | 25/7√2 |
| Rationalised denominator | 25√2/14 |
| Decimal form | 2.52538 |
Prime Factorisation
Numerator: 5^4
Denominator: 2 × 7^2
Perfect-Square Extraction
Numerator: 625 × 1
Denominator: 49 × 2
Step-by-Step Solution
- Start with √(625/98).
- The fraction is already in lowest terms.
- Extract square factors: √625 = 25.
- Extract square factors: √98 = 7√2.
- Write the extracted form as 25/7√2.
- Multiply by √2/√2 to rationalise the denominator.
- Simplify the coefficients and radical to obtain 25√2/14.
- Round the decimal value to 5 places: 2.52538.
Complete Working
Expression: √(625/98) Exact result: 25√2/14 Rationalised result: 25√2/14 Decimal: 2.52538 Approximation error: 1.36138052653e-6 Steps: - Start with √(625/98). - The fraction is already in lowest terms. - Extract square factors: √625 = 25. - Extract square factors: √98 = 7√2. - Write the extracted form as 25/7√2. - Multiply by √2/√2 to rationalise the denominator. - Simplify the coefficients and radical to obtain 25√2/14. - Round the decimal value to 5 places: 2.52538.
Calculator Inputs
Example Inputs
| Input | Simplified exact form | Approximate value |
|---|---|---|
| √(625/98) | 25√2/14 | 2.52538 |
| √(72/50) | 6/5 | 1.20000 |
| √(45/8) | 3√10/4 | 2.37171 |
| √(144/25) | 12/5 | 2.40000 |
Formula Used
√(a/b) = √a / √b √(a/b) = (√a × √b) / b √(625/98) = 25√2/14
Split the radical across numerator and denominator. Extract every available perfect-square factor. Rationalising removes radicals from the denominator for cleaner results.
How to Use
- Select fraction, whole-number, or expression input mode.
- Enter positive integer values in the visible fields.
- Choose rounding, factorisation, and rationalisation options.
- Press the simplify button to view exact steps.
- Copy, print, export, or save the completed result.
Common Mistakes
Do not add numerator and denominator before simplifying. Never leave zero inside the denominator field. Check extracted factors before rationalising the remaining denominator radical.
- Forgetting to reduce a common fraction factor.
- Extracting a factor that is not a perfect square.
- Changing only one side during rationalisation.
- Using a rounded decimal as the exact answer.
Calculation History
| Time | Expression | Exact result | Decimal |
|---|---|---|---|
| No saved calculations yet. | |||
Frequently Asked Questions
What is the simplified square root of 625/98?
The exact rationalised form is 25√2/14. Its decimal value is approximately 2.52538. Both forms represent precisely the same positive number.
Why does √625 become 25?
The number 625 equals 25 multiplied by 25. Therefore, 625 is a perfect square. Its principal square root is the positive value 25.
How does √98 simplify?
Factor 98 as 49 multiplied by 2. The square root of 49 equals seven. Therefore, √98 simplifies directly to the expression 7√2.
Why rationalise the denominator?
Rationalising removes square-root terms from the denominator. This creates a conventional exact fractional form. It also makes equivalent expressions easier to compare accurately.
Can the calculator simplify other fractions?
Yes, enter any supported positive integer fraction. The calculator extracts squares and reduces coefficients. It then presents exact, rationalised, and decimal results clearly.
Can I keep an irrational denominator?
Yes, clear the rationalisation option before calculating. The extracted denominator radical will remain visible. The rationalised equivalent still appears for useful comparison.
What does approximation error mean?
Approximation error measures lost value after decimal rounding. More decimal places normally reduce that difference. Exact radical forms avoid rounding error entirely during calculations.
Does the calculator accept decimal inputs?
This version accepts positive integers for exact factorisation. Convert terminating decimals into fractions before submitting. That preserves exact steps and reliable radical simplification results.
Are copied and downloaded results exact?
Copied working includes exact and approximate forms. PDF output also records every displayed calculation step. CSV history stores expressions, results, decimals, and timestamps securely.