Square Root of 85 — Value, Method & How to Find

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.

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.

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)

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).

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