√ Square Root of 691 = 26.287
The square root of 691 is a value that when multiplied by itself gives 691. Learn 5 different methods to calculate √691 with step-by-step explanations.
What Is √691 and Why It Matters
The square root of 691 is 26.287 because 26.287 × 26.287 ≈ 691. 691 is not a perfect square, so its square root is an irrational number that continues forever without repeating. The simplified radical form is √691.
Understanding √691 helps build a strong foundation in algebra, geometry, and advanced mathematics. Whether you're a student learning square roots, a teacher preparing lessons, or a professional needing quick math references, this guide covers everything you need to know.
5 Ways to Calculate √691
Choose the method that works best for you, from simple mental math to advanced approximation techniques.
Mental Math (Multiplication)
Find the closest perfect squares: 26² = 676 and 27² = 729. Since 691 is between these, estimate around 26.287.
Prime Factorization
Prime factors: 691 = 691. Simplify: √691 = √691.
Long Division
Pair digits from right: 691 → 691. Continue with decimal pairs. Answer: 26.286879
Newton-Raphson (Iterative)
Formula: x₁ = (x₀ + n/x₀)/2. Starting with 26: iterations converge to 26.287.
Scientific Calculator
Press √, then 691, then = or Enter. Result: 26.286879.
Method Comparison
| Method | Best For | Result | Difficulty |
|---|---|---|---|
| Mental Math | Quick estimates | 26.287 | Easy |
| Prime Factorization | Simplifying radicals | √691 | Medium |
| Long Division | Exact decimals | 26.286879 | Hard |
| Newton-Raphson | Programming | 26.287 | Hard |
| Calculator | Everyday use | 26.286879 | Easy |
Real-World Uses of √691
Geometry & Area
A square with area 691 cm² has side length 26.287 cm. This is a classic example used in math classes worldwide.
Construction & Architecture
Builders and architects use √691 to calculate diagonal measurements, ensure right angles, and determine material quantities.
Statistics & Data Science
Standard deviation, variance, and other statistical measures frequently use square root calculations for data analysis.
Physics & Engineering
Square roots are essential in calculating velocity, acceleration, electrical resistance, and many scientific formulas.
Continue Your Learning Journey
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Frequently Asked Questions
Practice Problems
Final Thoughts
The square root of 691 is 26.287. Since 26.287 × 26.287 ≈ 691, this is an irrational number with infinite decimal places.
We've explored 5 different methods to calculate √691, from simple mental math to advanced iterative techniques. Each method offers a unique approach to understanding this fundamental mathematical concept.
"Mastering square roots opens the door to advanced mathematics and real-world problem solving."