Recent Calculations
How This Works
Pick your operation (square root, cube root, whatever), type your number, hit Calculate. Or just press Enter - keyboard shortcuts work.
Perfect squares like √25 give you exact answers (5). Everything else shows decimals plus the simplified radical form. So √50 becomes 5√2 ≈ 7.071.
You also get prime factorization, step-by-step breakdowns, and verification. Basically everything except your homework done for you.
Square Roots Explained
A square root is whatever you multiply by itself to get the original number. √25 = 5 because 5 × 5 = 25. That's the whole concept. Simple as that.
They're called "square" roots because of geometry - a square with 25 square units of area has sides that are √25 = 5 units long. You're finding the root (side length) of the square (area).
Squaring and taking square roots undo each other. Square the 5, get 25. Take √25, back to 5. Every positive number technically has two square roots (+5 and -5 both work), but the √ symbol always means positive.
Perfect Squares
Numbers like 1, 4, 9, 16, 25, 36... where the answer is a whole number. √16 = 4 exactly. But √17? That's about 4.123, an irrational number that continues forever. Most numbers aren't perfect squares.
The √ Symbol
Called a radical sign. The number underneath is the radicand. That small number above it (the index) tells you which root - ²√ for square (usually omitted), ³√ for cube, ⁴√ for fourth, and so on.
Other Roots
Cube root (∛x) finds what cubed equals x. So ∛8 = 2 because 2³ = 8. Fourth root, fifth root - same idea, different exponent.
What About Negatives?
Can't do it with real numbers. No real number times itself gives a negative result. Math has imaginary numbers to handle this (i = √(-1)), but most everyday calculations avoid that mess.
Calculating Them Yourself
First, memorize the common ones. You'll use these constantly:
See 144? That's √144 = 12. See 625? √625 = 25. Knowing these saves you from reaching for a calculator every time.
Quick Estimation
For √50, find the perfect squares on each side. 7² = 49 and 8² = 64, so √50 is between 7 and 8. Since 50 is way closer to 49, the answer's around 7.1 (actually 7.071).
Want more precision? Use proportions: (50-49)/(64-49) = 1/15, so √50 ≈ 7 + 1/15 ≈ 7.07. Close enough for most purposes.
Simplifying with Prime Factors
Break the number into prime factors, group them in pairs, pull the pairs out.
Example - √72:
- Factor: 72 = 2 × 2 × 2 × 3 × 3 = 2³ × 3²
- Group: (2² × 3²) × 2
- Pull out pairs: 2 × 3 × √2 = 6√2
- Decimal: 6 × 1.414... ≈ 8.485
How Computers Actually Do It
Newton's method. Start with a guess, then repeatedly apply x_next = (x + n/x) / 2 until the answer stops changing. Turns out this converges stupid fast - usually done in 5-6 iterations.
Simplifying Radicals
√50 and 5√2 both equal 7.071..., but 5√2 is the "simplified" form. Why bother? Because it's cleaner for algebra and you can actually add/subtract radicals in simplified form.
Here's the thing - a simplified radical has no perfect square factors under the √. So √50 needs simplifying because 50 = 25 × 2, and 25 is a perfect square.
The Process
Find the biggest perfect square that divides your number. Pull it out using √(a × b) = √a × √b.
√48 becomes √(16 × 3) = √16 × √3 = 4√3.
√128 becomes √(64 × 2) = 8√2.
But √13 can't be simplified - 13 is prime. √21 can't either - it's 3 × 7 with no perfect square factors. Sometimes you're done already.
Testing Whether You're Done
Factor the radicand. Any prime that appears twice can come out. If all primes appear once, you're finished.
Perfect Squares and Cubes
Here's your reference list for perfect squares:
And perfect cubes:
Notice 64 appears in both - it's 8² and also 4³.
Quick pattern: squares only end in 0, 1, 4, 5, 6, or 9. Never 2, 3, 7, or 8. So if you see a number ending in 7, like 137, it can't be a perfect square. Simple test.
Also fun - the sum of the first n odd numbers always equals n². Try it: 1 = 1², 1+3 = 4 = 2², 1+3+5 = 9 = 3². Keeps working forever.
Where This Shows Up
So when do you actually need this stuff? Turns out, constantly if you're working with:
Right triangles and distance. The Pythagorean theorem (a² + b² = c²) needs square roots to find that third side. Building a 12×16 ft deck? The diagonal should measure exactly 20 ft (that's √400). GPS systems use the same math to calculate distances between coordinates.
Area problems. Got a square with 64 m² of area? Each side is √64 = 8 m. Works for circles too - if you know the area, you can work backward to find the radius.
Physics formulas. Kinetic energy, free fall calculations, and pendulum periods all involve square roots. Also voltage - your "120V" household AC actually swings from +170V to -170V, but the RMS (root mean square) value is 120V.
Statistics and finance. Standard deviation uses square roots to convert variance back to useful units. And here's a weird one - volatility scales with the square root of time, not linearly. So daily volatility of 2% becomes 2% × √252 ≈ 31.7% annually.
3D graphics, machine learning, signal processing - they all use them too. But you get the idea.
Common Screwups
Quick shortcut: use nearby perfect squares for estimates. √90 sits between √81=9 and √100=10, so it's about 9.5 (actually 9.49).
Another trick: √(4x) = 2√x. So √400 = √(4×100) = 2√100 = 20.
Things That Don't Work
You CANNOT split addition: √(9+16) ≠ √9 + √16. Left side is √25 = 5. Right side is 3+4 = 7. They're not equal. But multiplication? That splits fine - √(a×b) = √a × √b.
√(x²) equals |x|, the absolute value. Not x. So √((-5)²) = √25 = 5, not -5. The √ symbol means positive only.
And always simplify your final answer. Write √50 as 5√2.
Don't confuse (√a)² with √(a²). First one: (√5)² = 5. Second one: √(5²) = 5 and √((-5)²) = 5. The √ kills the sign.
With fractions: √(1/4) = √1/√4 = 1/2. Take the root of top and bottom separately.
Common Questions
What is a square root? The number that multiplied by itself gives you the original. So √25 = 5 because 5×5 = 25.
Can you take the square root of negatives? Not with real numbers. You need imaginary numbers for that (where i = √(-1)). But most practical calculations stick to positive numbers anyway.
Why simplify radicals? Because 5√2 is way easier to work with than √50, even though they're the same number. Plus you need simplified form to add or subtract radicals.
What are perfect squares? Numbers like 1, 4, 9, 16, 25, 36... where the square root is a whole number. They come from squaring integers.
How do calculators do it so fast? Newton's method - they make a guess, then keep improving it with a formula until it's accurate enough. Takes maybe 5-6 iterations tops.