Simplify the Square Root of 43681 = 209
Quick Answer
√43681 = 209
Formula: 43681 = 2092 → √43681 = 209
Simplify Square Root Calculator
Use the calculator above to simplify any square root to its simplest radical form (e.g. √50 = 5√2). See below the step-by-step method using prime factorization, plus a table of common simplified square roots.
43681 Is a Perfect Square
43681 = 209², so √43681 = 209.
Formula
To rewrite any square root in simplest radical form:
√(a2 · b) = a · √b
Where:
- a2 = the largest perfect-square factor of the radicand (the number under the √ symbol)
- b = the remaining squarefree factor (no perfect square divides it other than 1)
Worked example for √72:
- 72 = 36 × 2 = 62 × 2
- So a2 = 36, a = 6, b = 2
- Therefore √72 = 6√2
How to Simplify Square Roots
Simplifying a square root means rewriting it in the simplest radical form, where the number under the radical (the radicand) has no perfect square factors other than 1.
Method: Prime Factorization
- Find the prime factorization of the radicand.
- Identify pairs of identical prime factors.
- Each pair contributes one factor outside the radical.
- Any unpaired prime factor stays under the radical.
Example: Simplify √72
- 72 = 2 × 2 × 2 × 3 × 3 = 23 × 32
- We have a pair of 2's, a pair of 3's, and a leftover 2.
- Pair of 2's → contributes one 2 outside. Pair of 3's → contributes one 3 outside. Leftover 2 stays inside.
- √72 = (2 × 3)√2 = 6√2
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Want the Decimal Value Instead?
If you want the decimal approximation (e.g. √50 ≈ 7.0711) with the Babylonian Method step-by-step, use our Square Root Calculator.
Is 43681 Itself a Perfect Square?
If the radicand 43681 is itself a perfect square (e.g. 144 = 12²), then √43681 simplifies to an integer. Check with our Perfect Square Checker for the verdict + prime-factorization proof.
References
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