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# Square Root of 50 — Value, Radical Form, Methods

## TL;DR  
The square root of 50 is 5\sqrt{2} in simplified radical form and approximately 7.0711 as a decimal — irrational, because the prime 2 inside the radical has no partner. This article covers the exact value, three methods to compute it (prime factorization, repeated subtraction, long division), where \sqrt{50} shows up in geometry, and the slips students make most often.

**Result:** \sqrt{50} = 5\sqrt{2} \approx 7.0711

**Notation:** Simplified radical form 5\sqrt{2}; decimal approximation 7.0711 (4 d.p.).

**Method shown:** Prime factorization (Quick), repeated subtraction (Standard, with a Wrong-Path-First detour), long division (Stretch).

**Approximate value:** 7.071067811877 (11 d.p.).

**Exact form:** 5\sqrt{2} — cannot be reduced further, since 2 is prime.

## Quick Reference Table — Square Roots Near 50

| n | n\sqrt{n} (exact form) | n\sqrt{n} (4 d.p.) | Nearest perfect square |  
|---|---|---|---|  
| 45 | 3\sqrt{5} | 6.7082 | 49 |  
| 48 | 4\sqrt{3} | 6.9282 | 49 |  
| 49 | 7 | 7.0000 | 49 (itself) |  
| **50** | **5\sqrt{2}** | **7.0711** | **49** |  
| 52 | 2\sqrt{13} | 7.2111 | 49 |  
| 54 | 3\sqrt{6} | 7.3485 | 49 |  
| 56 | 2\sqrt{14} | 7.4833 | 49 |  
| 60 | 2\sqrt{15} | 7.7460 | 64 |  
| 64 | 8 | 8.0000 | 64 (itself) |

\sqrt{50} sits between 49=7 and 64=8 — closer to 7 because 50 is one away from 49, and 14 away from 64.

## Where \sqrt{50} shows up  
\sqrt{50} is the diagonal of a 5×5 square — by Pythagoras, \sqrt{5^2 + 5^2} = \sqrt{50} = 5\sqrt{2}. The same value appears as the hypotenuse of any right triangle with two legs of length 5, and as the distance between the points (0,0) and (5,5) on a coordinate plane. In physics, \sqrt{50} shows up in the magnitude of a vector with equal x and y components of 5 units.

### What "square root of 50" means  
The square root of a non-negative number n is the value x such that x² = n. For \sqrt{50}, the positive x with x² = 50. Because 7² = 49 and 8² = 64, \sqrt{50} lives between 7 and 8.

The simplified-radical form 5\sqrt{2} extracts the part of 50 that is a perfect square (25) and leaves the rest under the radical. The decimal form 7.0711 comes from numerically approximating the irrational tail.

## Is \sqrt{50} rational or irrational?  
\sqrt{50} is **irrational**. Reason: a number is a perfect square if and only if every prime in its factorization appears to an _even_ power. 50 = 2ⁱ⋅5² — the 5 pairs up neatly into 5², but the 2 stands alone with exponent 1. That lone 2 is the reason \sqrt{50} doesn't collapse to an integer.

The decimal 7.0710678118… neither terminates nor repeats. Like every irrational, it can be approximated but never written out exactly in decimal form.

## How to find \sqrt{50} — three methods

### Method 1 — Prime factorization (Quick)  
Break 50 into prime factors, then pair them up.

50 = 2⋅5⋅5 = 2⋅5²

Pull the pair of 5s out of the radical:  
\sqrt{50} = \sqrt{2 ⋅ 5²} = 5\sqrt{2}

**Final answer:** \sqrt{50} = 5\sqrt{2} \approx 7.0711.

### Method 2 — Repeated subtraction (Standard, Wrong-Path-First)  
A common student instinct is to try subtracting odd numbers from 50. Start at 50. Subtract consecutive odd numbers:  
50−1 = 49, 49−3 = 46, 46−5 = 41, 41−7 = 34, 34−9 = 25, 25−11 = 14, 14−13 = 1.  
After seven subtractions, the remainder is 1 — not 0.  
That non-zero remainder is the signal: 50 is not a perfect square. The rescue is prime factorization. 50 = 5²⋅2, so \sqrt{50} = 5\sqrt{2}.

**Final answer:** \sqrt{50} = 5\sqrt{2}.

### Method 3 — Long division (Stretch)  
For a decimal approximation to four places, run the long-division algorithm on 50.000000.

**Step 1.** Largest integer with square ≤50 is 7 (7² = 49). Subtract: 50−49 = 1. Bring down 000000: 100.

**Step 2.** Double 7: 14. Find d with (14+d)⋅d≤100. d=0 gives 14⋅0=0. Subtract: 100−0=100. Bring down 000000: 10,000.

**Step 3.** Double 7: 14. Find d with (140+d)⋅d≤10,000. d=7 gives 140⋅7=980. Subtract: 10,000−980=20. Bring down 000000: 20,000.

Continuing produces \sqrt{50} \approx 7.0711.

## Conclusion  
- The **square root of 50** simplifies to 5\sqrt{2} — exact form — and approximates to 7.0711 as a decimal.  
- \sqrt{50} is irrational; the prime factor 2 inside the radical has no partner.  
- Three methods reach the answer: prime factorization (one line), repeated subtraction (signals the irrationality), long division (gives the decimal).  
- Only perfect-square factors leave the radical, and they leave as their square root — not as themselves.