Square Root of 245 — Value, Simplification, Steps

Square Root of 245 — Value, Simplification, Steps

TL;DR

The square root of 245 simplifies to 7√5 ≈ 15.652, because 245 factors as 5 × 7² and the perfect square 49 pulls out as 7. This article gives the value, the full simplification, two computation methods, and the mistakes students make with √245.

The square root of 245 is approximately 15.652, and in exact form it is 7√5. The 245 hides a perfect square — a factor of 49 — which comes out of the radical as 7 and leaves a 5 behind.

Quick Answer:

Result: 245=75≈15.652\sqrt{245} = 7\sqrt{5} \approx 15.652245​=75​≈15.652

Notation: Simplified radical 757\sqrt{5}75​; decimal 15.652515.652515.6525 (to 4 dp)

Method shown: Prime factorisation to extract the perfect-square factor, then estimation

Approximate value: 15.6525 (irrational, non-terminating)

Exact form: 757\sqrt{5}75​

Quick Reference Table of Nearby Square Roots

The table places √245 among nearby values so the simplification and the size both make sense at a glance.

Number nnn Square root n\sqrt{n}n​ Simplified form
225 225=15\sqrt{225} = 15225​=15 Exact (perfect square)
245 245≈15.652\sqrt{245} \approx 15.652245​≈15.652 757\sqrt{5}75​
256 256=16\sqrt{256} = 16256​=16 Exact (perfect square)
125 125≈11.180\sqrt{125} \approx 11.180125​≈11.180 555\sqrt{5}55​
500 500≈22.361\sqrt{500} \approx 22.361500​≈22.361 10510\sqrt{5}105​
5 5≈2.236\sqrt{5} \approx 2.2365​≈2.236 5\sqrt{5}5​

Every value in the last two rows shares the same √5 core, which is exactly why √245 = 7√5 lines up so neatly with them.

Where the Square Root of 245 Appears

A square root answers "what side gives this area?" So √245 is the side of a square whose area is 245 square units. It also turns up through the Pythagorean theorem: a right triangle with legs 14 and 7 has a hypotenuse of 142+72=196+49=245=75\sqrt{14^2 + 7^2} = \sqrt{196 + 49} = \sqrt{245} = 7\sqrt{5}142+72​=196+49​=245​=75​. Any distance or diagonal that resolves to 245 under the root carries this same value.

What a Square Root Means

The square root of a number nnn is the value that, multiplied by itself, gives nnn. In symbols, n=x\sqrt{n} = xn​=x means x2=nx^2 = nx2=n. When nnn is a perfect square the answer is a whole number; when it is not, the root is irrational and its decimal runs on forever without repeating.

245 is not a perfect square, so √245 is irrational. But it is not fully "stuck" either — part of it simplifies, because 245 contains the perfect square 49.

How to Compute the Square Root of 245

Method 1: Prime factorisation (the simplification)

Break 245 into primes and look for pairs.

245=5×49245 = 5 \times 49245=5×49

245=5×7×7245 = 5 \times 7 \times 7245=5×7×7

245=5×72245 = 5 \times 7^2245=5×72

A pair of identical primes (7×77 \times 77×7) leaves the radical as a single 7. The lone 5 stays inside.

245=72×5\sqrt{245} = \sqrt{7^2 \times 5}245​=72×5​

245=75\sqrt{245} = 7\sqrt{5}245​=75​.

Final answer: 245=75\sqrt{245} = 7\sqrt{5}245​=75​.

Method 2: Estimation by bracketing

Trap √245 between two perfect squares.

152=22515^2 = 225152=225

162=25616^2 = 256162=256

So 15<245<1615 < \sqrt{245} < 1615<245​<16. Because 245 is much closer to 256 than to 225, the answer is close to 15.7.

15.62=243.3615.6^2 = 243.3615.62=243.36

15.72=246.4915.7^2 = 246.4915.72=246.49

Final answer: 245≈15.652\sqrt{245} \approx 15.652245​≈15.652, matching 75=7×2.2361=15.65257\sqrt{5} = 7 \times 2.2361 = 15.652575​=7×2.2361=15.6525.

Common Mistakes With Square Root of 245

Mistake 1: Missing the perfect-square factor

Where it slips in: stopping at "245 is not a perfect square, so it can't be simplified."

Don't do this: leave the answer as a bare 245\sqrt{245}245​ when a factor of 49 is waiting inside.

The correct way: always factor fully. Students first simplifying radicals often check only whether the whole number is a perfect square and forget to look for a perfect-square factor. Here 245=49×5245 = 49 \times 5245=49×5, so 245=75\sqrt{245} = 7\sqrt{5}245​=75​.

Mistake 2: Pulling out the wrong number

Where it slips in: knowing 49 comes out, but writing the 49 instead of its root.

Don't do this: write 245=495\sqrt{245} = 49\sqrt{5}245​=495​.

The correct way: the factor 49 leaves the radical as 49=7\sqrt{49} = 749​=7, not as 49. The result is 757\sqrt{5}75​.

Mistake 3: Splitting the sum under the root

Where it slips in: using √245 inside a Pythagoras step.

Don't do this: claim 196+49=196+49=14+7=21\sqrt{196 + 49} = \sqrt{196} + \sqrt{49} = 14 + 7 = 21196+49​=196​+49​=14+7=21.

The correct way: the root of a sum is not the sum of the roots. Add first, then take the root: 196+49=245=75≈15.652\sqrt{196 + 49} = \sqrt{245} = 7\sqrt{5} \approx 15.652196+49​=245​=75​≈15.652.

Conclusion