Square Root Calculator
Calculate principal square roots (√x), simplify radicals into a√b form, check perfect squares, and view positive/negative real roots.
√x Square Root Solver
Enter non-negative radicand (x ≥ 0).
Square Root Output
Quick Summary
Our square root calculator radical solver steps tool computes principal non-negative square roots ($\sqrt{x}$), simplifies radical expressions into $a\sqrt{b}$ simplified form, verifies perfect square integers, and displays both real solutions ($\pm \sqrt{x}$).
How It Works: Radical Simplification & Roots
A square root of a number $x$ is a number $y$ such that $y^2 = x$:
1. **Principal Square Root ($\sqrt{x}$):** The non-negative solution to $y^2 = x$ (e.g. $\sqrt{25} = 5$).
2. **Both Real Roots ($\pm \sqrt{x}$):** Every positive number has two real square roots ($+5$ and $-5$ because $(-5)^2 = 25$).
3. **Simplifying Radicals ($a\sqrt{b}$):** Factor out perfect square factors from the radicand: $\sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2}$.
4. **Perfect Squares:** Whole numbers whose square roots are integers ($1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, \dots$).
Formula Explanation
Your square root calculations follow standard radical equations:
Step-by-Step Worked Example
Here is a detailed 5-step breakdown for simplifying √50:
- Step 1 (Identify Radicand): Radicand input $x = 50$.
- Step 2 (Find Prime Factorization): $50 = 2 \times 5 \times 5 = 2 \times 5^2$.
- Step 3 (Group Perfect Squares): $\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2}$.
- Step 4 (Extract Integer Root): $\sqrt{25} = 5 \implies \mathbf{5\sqrt{2}}$.
- Step 5 (Evaluate Decimal Approximation): $5 \times 1.414214 \approx \mathbf{7.071068}$, delivering **√50 = 5√2 ≈ 7.071068**!
Calculation Examples: Real-World Scenario Comparison
Compare principal square roots, simplified radicals, and perfect square verifications:
| Radicand (x) | Principal Root √x | Simplified Radical Form (a√b) | Perfect Square? | Both Real Roots (±√x) |
|---|---|---|---|---|
| 144 | 12.00 | 12 (Exact Integer) | YES (12² = 144) | ±12 |
| 50 | 7.071068 | 5√2 | NO (Irrational) | ±7.071068 |
| 0.25 | 0.50 | 1/2 (Decimal 0.5) | NO (Rational Decimal) | ±0.5 |
| 2 | 1.414214 | √2 (Irrational Constant) | NO (Irrational) | ±1.414214 |
Benefits of Using the Square Root Calculator
Utilizing this calculator provides essential Pythagorean geometry, physics velocity, and statistics advantages:
- Pythagorean Theorem Geometry ($c = \sqrt{a^2 + b^2}$): Calculates exact hypotenuse lengths for right triangles.
- Radical Simplification Engine ($a\sqrt{b}$): Automatically factors out perfect squares for algebra homework and test prep.
- Standard Deviation & Statistics ($s = \sqrt{\sigma^2}$): Computes standard deviation from sample variance.
- Physics Free Fall & Kinetic Energy ($v = \sqrt{2gh}$): Solves velocity from height under gravitational acceleration.
Frequently Asked Questions (FAQ)
What is a square root?
A square root of a number x is a value y such that y × y = x (e.g. 5 is a square root of 25 because 5 × 5 = 25).
What is the principal square root?
The principal square root is the positive (non-negative) square root of a number, denoted by the radical symbol √x.
Why does a positive number have two square roots?
Because both (+y)^2 and (-y)^2 equal positive x (e.g. (+5)^2 = 25 and (-5)^2 = 25).
How do you simplify a radical expression like √50?
Factor out the largest perfect square: √50 = √(25 × 2) = √25 × √2 = 5√2.
What is a perfect square?
A perfect square is an integer that is the square of an integer (e.g. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144).
What is the square root of a negative number?
In real numbers, the square root of a negative number is undefined. In complex numbers, it uses the imaginary unit i (√(-x) = i√x).
What is √0?
√0 = 0 (since 0 × 0 = 0).
What is √2?
√2 ≈ 1.41421356 (an irrational number that cannot be written as an exact fraction).
What is √3?
√3 ≈ 1.7320508.
Is √(a + b) equal to √a + √b?
NO! √(a + b) is NOT equal to √a + √b (e.g. √(9+16) = √25 = 5, but √9 + √16 = 3 + 4 = 7).