# Square Root of 85 — Value, Method & How to Find  
### TL;DR  
The square root of 85 is approximately \(85 \approx 9.2195\) — irrational, non-terminating, and cannot be simplified into a cleaner radical form. This article covers the value in exact and decimal form, three methods to compute it, where \(85\) shows up, and the slips students make most often.  
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### The Answer At A Glance  
**Result:** \(85 \approx 9.21954445729\)  
**Notation:** Decimal approximation; exact form is \(85\).  
**Method shown:** Long division method (manual), with cross-checks using Newton's method and linear interpolation.  
**Approximate value:** \(9.2195\) (4 d.p.)  
**Exact form:** \(85\) — cannot be simplified, since \(85=5 \times 17\) has no square factor.  
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### Quick Reference Table — Square Roots Near 85  
| n | \(n\sqrt{n}\) (exact) | \(n\sqrt{n}\) (4 d.p.) |  
|---|---|---|  
| 81 | \(9\) | \(9.0000\) |  
| 82 | \(\sqrt{82}\) | \(9.0554\) |  
| 83 | \(\sqrt{83}\) | \(9.1104\) |  
| 84 | \(\sqrt{84}\) | \(9.1652\) |  
| **85** | \(\sqrt{85}\) | **9.2195** |  
| 86 | \(\sqrt{86}\) | \(9.2736\) |  
| 87 | \(\sqrt{87}\) | \(9.3274\) |  
| 88 | \(\sqrt{88}\) | \(9.3808\) |  
| 89 | \(\sqrt{89}\) | \(9.4340\) |  
| 90 | \(\sqrt{90}\) | \(9.4868\) |  
| 100 | \(10\) | \(10.0000\) |  
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Between \(81=9\) and \(100=10\), \(85\) sits closer to the lower end at \(9.22\). Only \(81\) and \(100\) in this band have integer square roots.  
### What "Square Root of 85" Means  
The square root of a non-negative number \(n\) is the value \(x\) such that \(x^2=n\). For \(85\), that means the positive number \(x\) for which \(x \cdot x=85\).  
Because \(9^2=81\) and \(10^2=100\), \(85\) falls between \(9\) and \(10\) — closer to \(9\) than to \(10\).  
### Is \(\sqrt{85}\) Rational or Irrational?  
\(\sqrt{85}\) is **irrational**. Reason: \(85=5\times17\). Neither \(5\) nor \(17\) is a perfect square. A number is a perfect square if and only if every prime in its factorization appears to an even power. Since both \(5\) and \(17\) appear to the first power, \(85\) is not a perfect square, and its square root cannot be written as a fraction \(\frac{p}{q}\).  
The decimal \(9.21954445729…\) neither terminates nor repeats — the definition of irrational.  
### How To Find \(\sqrt{85}\) — Three Methods  
#### Method 1 — Long division (digit by digit)  
Pair the digits of 85: \(85.000000\).  
1. Largest integer with square \(\leq 85\) is \(9\) (\(9^2 = 81\)). Subtract: \(85−81=4\). Bring down \(000000\): \(400\).  
2. Double \(9\): \(18\). Find \(d\) with \((180+d)\cdot d\leq 400\). \(d=2\) gives \(182\cdot2=364\). Subtract: \(400−364=36\). Bring down \(000000\): \(3600\).  
3. Double \(9.2\): \(18.4\). Find \(d\) with \((1840+d)\cdot d\leq 3600\). \(d=1\) gives \(1841\). Subtract: \(3600−1841=1759\). Bring down \(000000\): \(175900\).  
4. Double \(9.219\): \(18.438\). \(d=9\) gives \(18429\cdot9=165861\). Subtract.  
After four steps: \(85 \approx 9.219\). Continuing gives \(9.2195\).  
**Final answer:** \(85 \approx 9.2195\).  
#### Method 2 — Newton's iteration (fastest)  
\(x_{k+1}=\frac{1}{2}(x_k + \frac{n}{x_k})\)  
Start \(x_0=9\).  
- \(x_1=\frac{1}{2}(9 + \frac{85}{9})=9.2222\)  
- \(x_2=\frac{1}{2}(9.2222 + \frac{85}{9.2222})=9.2195\)  
Two iterations to four-decimal precision.  
#### Method 3 — Linear interpolation (mental estimate)  
\(85 \approx 9 + \frac{85 - 81}{100 - 81} = 9 + \frac{4}{19} \approx 9.218\)  
Quick enough for a sanity check.  
### Where \(\sqrt{85}\) Shows Up  
\(\sqrt{85}\) appears as the diagonal of a 2×9 rectangle. It also appears in the distance between (0,0) and (6,7): \(\sqrt{36 + 49} = \sqrt{85}\).  
### Three Slips That Cost Marks on \(\sqrt{85}\)  
#### **Mistake 1: Trying to simplify when no square factor exists.**  
**Where it slips in:** Students assume every non-perfect-square integer simplifies to \(a\sqrt{b}\).  
**Don't do this:** \(85=5⋅17\) → "simplifies to" something cleaner.  
**The correct way:** Check whether the radicand has a perfect-square factor other than 1.  \(85\) is already in simplest radical form.  
#### **Mistake 2: Reporting a truncated decimal as exact.**  
**Don't do this:** \(85 \approx 9.2195\).  
**The correct way:** Keep the exact value as \(85\).  
#### **Mistake 3: Confusing \(\sqrt{85}\) with \(±\sqrt{85}\).**  
**Where it slips in:** Solving \(x^2=85\) only results in the positive root.  
**The correct way:** Both positive and negative roots exist, therefore \(x = ±\sqrt{85}\).  
### Conclusion  
- The **square root of 85** is approximately 9.2195 — irrational, non-terminating, non-repeating.  
- \(85=5×17\) has no perfect-square factor, so it cannot be simplified.  
- Three methods to compute it: long division, Newton's iteration, and linear interpolation.  
- Always use \(≈\) for irrational decimal approximations.  
- \(\sqrt{85}\) shows up as the diagonal of a 2×9 rectangle and in standard distance-formula calculations.