Calculator Tools
Square Root Calculator
Find the square root of any number with decimal, simplified radical, prime factorization, perfect square check, and step-by-step work.
Square root calculator
Enter a number to find its square root
Integers, decimals (12.25), fractions (9/4), scientific notation (1.5e3), and negatives are all accepted.
Principal square root
8.4852813742
72 is not a perfect square
Positive root
+8.4852813742
Negative root
-8.4852813742
Scientific notation
8.485281e+0
Exact and simplified form
The form keeps the answer exact when possible: as a simplified radical for integers, as a clean decimal for fractions of the right shape, or as an imaginary expression for negatives.
Exact form
6*sqrt(2)
Simplified radical
6 * sqrt(2)
Prime factorization
72 = 2^3 * 3^2
Perfect square check
72 is not a perfect square
Step-by-step
Each step shows the rewrite or substitution used to reach the final form.
- Start with the input value 72.
- Find the prime factorization of 72: 72 = 2^3 * 3^2.
- Pair up matching prime factors. Each pair contributes its prime to the outside of the radical: sqrt(72) = 6*sqrt(2).
- Compute the decimal value: sqrt(72) ~ 8.4852813742.
How simplification works
- Factor the radicand into primes. For example, 72 = 2 * 2 * 2 * 3 * 3.
- Group the primes into pairs. Each pair of the same prime becomes a single copy outside the radical.
- Leftover primes stay inside the radical. The simplified form for 72 is 6 * sqrt(2), because the pair 2 * 2 and the pair 3 * 3 contribute 2 and 3, and the lone 2 stays inside.
- When every prime pairs up evenly, the radicand is a perfect square and the simplified form is a plain integer.
Why there are two roots
- Squaring a positive number and squaring its negative both give the same result, so x^2 = n has two real solutions whenever n is positive: +sqrt(n) and -sqrt(n).
- The principal square root is the non-negative one, written sqrt(n). When you only need the magnitude, that is the value most calculators return.
- For a negative radicand there is no real solution. The principal square root is defined as i * sqrt(|n|), where i is the imaginary unit with i^2 = -1.
- The square root of zero is just zero, with no separate negative root.
How to use
- Type a number into the input field. Integers, decimals, fractions, scientific notation, and negatives are all accepted.
- Read the principal square root at the top, alongside the negative root and a scientific notation form.
- Check the Exact and simplified form panel for the simplified radical (like 6*sqrt(2)), the perfect square check, and the prime factorization of integer inputs.
- For negative inputs read the imaginary result in the +/- a*i*sqrt(b) form.
- Open the Step-by-step pane to see the factoring, pairing, and extraction work used to reach the answer.
- Click Copy full solution to grab the input, both roots, exact form, factorization, and every step as plain text.
About this tool
Square Root Calculator finds the square root of any number you enter and shows the full picture, not just the decimal answer. The input field accepts integers, decimals (12.25), simple fractions (9/4), scientific notation (1.5e3), and negative numbers. The principal square root appears as a precise decimal, alongside its negative counterpart and a scientific notation form for very large or very small magnitudes. For non-negative integer inputs the tool factors the radicand into primes (for example 72 = 2^3 * 3^2), pairs the factors, and produces the exact simplified radical form (sqrt(72) = 6 * sqrt(2)). When every prime appears an even number of times, it flags the input as a perfect square and shows the integer root with a verification line. Negative inputs are reported as imaginary numbers in the i * sqrt(|n|) form, with the same simplification applied to the magnitude under the radical when possible. Fractions and certain decimals whose square root is itself a clean rational (like 0.25, 6.25, or 9/4) get an exact decimal form too. A step-by-step pane walks through the work the way a textbook would: factor, pair, extract, multiply, and the final decimal. Two reference panels explain how radical simplification works and why every positive number has both a positive and a negative square root. Copy buttons sit on every result row, plus a single Copy full solution button that grabs the input, both roots, exact form, complex form when applicable, prime factorization, and every step as plain text. All math runs in your browser. No values are uploaded.
Free to use. Works in your browser. No signup, no login.
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