Square Root Division Calculator

Compute √a ÷ √b or √(a/b) effortlessly today, fast. Get exact radicals, decimals, and rationalized denominators, instantly now. Simplify factors, extract squares, reduce fractions with explanations included. Share results, save scenarios, control rounding and precision. Download CSV, export PDF, and review clear step breakdowns.

Results
Input: √(50) ÷ √(2) Decimal: 5 Rationalized: 5
Key identities & simplification patterns
Identity / Pattern Conditions Example Result
√a ÷ √b = √(a/b) b > 0 √18 ÷ √8 3/4 × √2
Rationalize denominator b > 0 √7 ÷ √50 √14 / 10
Factor out squares a, b ≥ 0 √(k²m) = k√m k outside; m square‑free
Fractions inside radicals q, s ≠ 0 √(p/q) ÷ √(r/s) √(ps/qr)
Negatives introduce i Use principal root √(-a) ÷ √b i·√a / √b
Use these to quickly rewrite, cancel factors, and rationalize results.
Worked practice set
# A B Simplified / Rationalized Decimal (6 d.p.)
19887/23.5
212752/50.4
327333
445203/21.5
53/81/2√3 / 20.866025
6750√14 / 100.374166
These entries recompute numerically from the same engine used above.
Note: Symbolic simplification treats integer and fractional radicands exactly; decimals are approximated as rational numbers for improved simplification.
Steps
  1. Rewrite expression as √A ÷ √B. If desired, rationalize by multiplying numerator and denominator by √B.
  2. Separate numerators and denominators inside radicals: √(A) = √(num(A)) / √(den(A)), similarly for √(B).
  3. Collect outside coefficients from perfect square factors. Remaining factors stay inside radicals.
  4. Combine inside radicals: √(rN)/√(rD) with rN=2 and rD=2.
  5. Rationalize: multiply by √(rD)/√(rD) → numerator √(rN·rD), denominator rD. Simplify the new radical.
  6. Evaluate numerically using the chosen rounding mode and precision.
Key identities & simplification patterns
Identity / Pattern Conditions Example Result
√a ÷ √b = √(a/b) b > 0 √18 ÷ √8 3/4 × √2
Rationalize denominator b > 0 √7 ÷ √50 √14 / 10
Factor out squares a, b ≥ 0 √(k²m) = k√m k outside; m square‑free
Fractions inside radicals q, s ≠ 0 √(p/q) ÷ √(r/s) √(ps/qr)
Negatives introduce i Use principal root √(-a) ÷ √b i·√a / √b
Use these to quickly rewrite, cancel factors, and rationalize results.
Worked practice set
# A B Simplified / Rationalized Decimal (6 d.p.)
19887/23.5
212752/50.4
327333
445203/21.5
53/81/2√3 / 20.866025
6750√14 / 100.374166
These entries recompute numerically from the same engine used above.
Formula used
How to use this calculator
  1. Enter A and B as integers, fractions (e.g., 3/5), or decimals.
  2. Choose decimal places and a rounding mode for the numeric result.
  3. Toggle “Rationalize denominator” to remove radicals from the denominator.
  4. Press Calculate. Review the exact form, rationalized form, and decimal value.
  5. Use Download CSV or Download PDF to export your results.
Example data
# A B Exact / Rationalized Decimal
1 50 2 5 5
2 502 5 × √25
3 188 3/4 × √21.5
4 3/512 1/10 × √20.223607
5 2.250.09 55
6 -82 i × 22i
These examples demonstrate simplification, rationalization, fractional inputs, decimal conversion, and imaginary cases.
FAQs

For b > 0, the principal square‑root rule applies: √a ÷ √b = √(a/b). If b is negative, handle imaginary units carefully; if both are negative, i/i cancels.

Rationalizing removes radicals from denominators, producing forms like k·√m / n or simply k·√m. This often simplifies further algebra and matches conventional presentation standards.

Yes. The calculator converts decimals to rational numbers internally for better symbolic simplification. Fractions like 7/12 are handled exactly.

We use √(-x) = i√x. If both A and B are negative, the i factors cancel. If only one is negative, the result is purely imaginary.

Choose Round, Floor, Ceil, or Truncate. Precision is user‑selectable up to 12 decimal places.
Key identities & simplification patterns
Identity / Pattern Conditions Example Result
√a ÷ √b = √(a/b) b > 0 √18 ÷ √8 3/4 × √2
Rationalize denominator b > 0 √7 ÷ √50 √14 / 10
Factor out squares a, b ≥ 0 √(k²m) = k√m k outside; m square‑free
Fractions inside radicals q, s ≠ 0 √(p/q) ÷ √(r/s) √(ps/qr)
Negatives introduce i Use principal root √(-a) ÷ √b i·√a / √b
Use these to quickly rewrite, cancel factors, and rationalize results.
Worked practice set
# A B Simplified / Rationalized Decimal (6 d.p.)
19887/23.5
212752/50.4
327333
445203/21.5
53/81/2√3 / 20.866025
6750√14 / 100.374166
These entries recompute numerically from the same engine used above.

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