Input
Steps
Result Log
| # | Time | Input | Exact form | Decimal |
|---|
Example Data
try these| Fraction | Expected exact | Approx. |
|---|
Formula used
√(a/b) = √a / √b, b > 0, a ≥ 0
- Simplify the fraction: a/b → a′/b′ with gcd(a′, b′) = 1.
- Extract perfect-square factors: a′ = p²·r, b′ = q²·s where r,s are square‑free.
- Then √(a/b) = (p/q)·√(r/s) = (p·√(r·s)) / (q·s).
- Rationalize the denominator so radicals appear only in the numerator.
Edge cases: a = 0 gives 0. Negative a/b is not supported in reals.
How to use this calculator
- Enter integers for numerator and denominator above.
- Press Compute to see exact and decimal results.
- Review each step to understand reductions and rationalization.
- Use Download CSV or Download PDF to save the log.
- Click Copy shareable link to preserve inputs in a URL.
How to find the square root of a fraction
- Reduce the fraction a/b to lowest terms.
- Factor: write a = p²·r and b = q²·s with r,s square‑free.
- Apply property: √(a/b) = (p/q)·√(r/s).
- If s > 1, rationalize: (p/q)·√(r/s) = (p·√(r·s))/(q·s).
- Simplify any new perfect squares and reduce the outside fraction.
Worked example 1
√(45/20)
- Reduce: 45/20 → 9/4.
- Squares: 9 = 3², 4 = 2².
- Therefore √(9/4) = 3/2.
Worked example 2
√(12/7)
- Already reduced.
- 12 = 2²·3, 7 is square‑free.
- √(12/7) = (2√3)/√7 = (2√21)/7.
Tip: Denominator must be nonzero. Negative fractions are outside real numbers.
Square‑free factors and perfect squares (reference)
These help decide what comes outside the radical and what stays inside.
| n | Largest square factor | Square‑free part |
|---|---|---|
| 8 | 4 = 2² | 2 |
| 12 | 4 = 2² | 3 |
| 18 | 9 = 3² | 2 |
| 20 | 4 = 2² | 5 |
| 27 | 9 = 3² | 3 |
| 45 | 9 = 3² | 5 |
| 50 | 25 = 5² | 2 |
| 63 | 9 = 3² | 7 |
| 72 | 36 = 6² | 2 |
Rule: write n = k²·r with r square‑free. Then √n = k√r.
Rationalizing denominators: quick patterns
- √(a/b) = (√a·√b)/b, then simplify and reduce.
- If b is square‑free: √(a/b) = (√(ab))/b, then extract any new squares.
- If b is a perfect square m², the result is √a / m.
Examples
√(1/2) = √2/2, √(3/5) = √15/5, √(12/7) = (2√21)/7
This tool automates these patterns and shows intermediate steps.
Exact radicals vs decimals: choosing the right form
- Exact form keeps radicals (e.g., 2√21/7) for algebra, proofs, and symbolic work.
- Decimal form is better for engineering tolerances and numeric comparisons.
- The calculator prints about 12 significant digits for the decimal output.
- Rounding error bound after d digits is roughly ≤ 0.5×10−d.
Switch forms depending on context—symbolic steps or numeric decisions.