Square Root Division Calculator

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.)
1	98	8	7/2	3.5
2	12	75	2/5	0.4
3	27	3	3	3
4	45	20	3/2	1.5
5	3/8	1/2	√3 / 2	0.866025
6	7	50	√14 / 10	0.374166

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

Rewrite expression as √A ÷ √B. If desired, rationalize by multiplying numerator and denominator by √B.
Separate numerators and denominators inside radicals: √(A) = √(num(A)) / √(den(A)), similarly for √(B).
Collect outside coefficients from perfect square factors. Remaining factors stay inside radicals.
Combine inside radicals: √(rN)/√(rD) with rN=2 and rD=2.
Rationalize: multiply by √(rD)/√(rD) → numerator √(rN·rD), denominator rD. Simplify the new radical.
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.)
1	98	8	7/2	3.5
2	12	75	2/5	0.4
3	27	3	3	3
4	45	20	3/2	1.5
5	3/8	1/2	√3 / 2	0.866025
6	7	50	√14 / 10	0.374166

These entries recompute numerically from the same engine used above.

Formula used

Division rule: √a ÷ √b = √(a/b), for b > 0.
Rationalizing rule: √a / √b = (√a·√b) / b, for b > 0.
Simplification: √(k²·m) = k·√m, where m is square‑free.
Imaginary handling: √(-x) = i·√x; i/i cancels when both radicands are negative.
Domain: √b requires b ≠ 0; choose rationalization to remove radical from denominator.

How to use this calculator

Enter A and B as integers, fractions (e.g., 3/5), or decimals.
Choose decimal places and a rounding mode for the numeric result.
Toggle “Rationalize denominator” to remove radicals from the denominator.
Press Calculate. Review the exact form, rationalized form, and decimal value.
Use Download CSV or Download PDF to export your results.

Example data

#	A	B	Exact / Rationalized	Decimal
1	50	2	5	5
2	50	2	5 × √2	5
3	18	8	3/4 × √2	1.5
4	3/5	12	1/10 × √2	0.223607
5	2.25	0.09	5	5
6	-8	2	i × 2	Fatal error: Uncaught Error: Call to undefined function str_ends_with() in /home/u294241901/domains/codingace.net/public_html/maths/square_root_division.php:188 Stack trace: #0 /home/u294241901/domains/codingace.net/public_html/maths/square_root_division.php(704): round_with_mode('2i', 'round', 6) #1 {main} thrown in /home/u294241901/domains/codingace.net/public_html/maths/square_root_division.php on line 188