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: $\sqrt{245} = 7\sqrt{5} \approx 15.652$
Notation: Simplified radical $7\sqrt{5}$; decimal $15.6525$ (to 4 dp)
Method shown: Prime factorisation to extract the perfect-square factor, then estimation
Approximate value: 15.6525 (irrational, non-terminating)
Exact form: $7\sqrt{5}$
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 $n$ | Square root $\sqrt{n}$ | Simplified form |
|---|---|---|
225 | $\sqrt{225} = 15$ | Exact (perfect square) |
245 | $\sqrt{245} \approx 15.652$ | $7\sqrt{5}$ |
256 | $\sqrt{256} = 16$ | Exact (perfect square) |
125 | $\sqrt{125} \approx 11.180$ | $5\sqrt{5}$ |
500 | $\sqrt{500} \approx 22.361$ | $10\sqrt{5}$ |
5 | $\sqrt{5} \approx 2.236$ | $\sqrt{5}$ |
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 $\sqrt{14^2 + 7^2} = \sqrt{196 + 49} = \sqrt{245} = 7\sqrt{5}$. 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 $n$ is the value that, multiplied by itself, gives $n$. In symbols, $\sqrt{n} = x$ means $x^2 = n$. When $n$ 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.
$245 = 5 \times 49$
$245 = 5 \times 7 \times 7$
$245 = 5 \times 7^2$
A pair of identical primes ($7 \times 7$) leaves the radical as a single 7. The lone 5 stays inside.
$\sqrt{245} = \sqrt{7^2 \times 5}$
$\sqrt{245} = 7\sqrt{5}$
Final answer: $\sqrt{245} = 7\sqrt{5}$.
Method 2: Estimation by bracketing
Trap √245 between two perfect squares.
$15^2 = 225$
$16^2 = 256$
So $15 < \sqrt{245} < 16$. Because 245 is much closer to 256 than to 225, the answer is close to 15.7.
$15.6^2 = 243.36$
$15.7^2 = 246.49$
Final answer: $\sqrt{245} \approx 15.652$, matching $7\sqrt{5} = 7 \times 2.2361 = 15.6525$.
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 $\sqrt{245}$ 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 $245 = 49 \times 5$, so $\sqrt{245} = 7\sqrt{5}$.
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 $\sqrt{245} = 49\sqrt{5}$.
The correct way: the factor 49 leaves the radical as $\sqrt{49} = 7$, not as 49. The result is $7\sqrt{5}$.
Mistake 3: Splitting the sum under the root
Where it slips in: using √245 inside a Pythagoras step.
Don't do this: claim $\sqrt{196 + 49} = \sqrt{196} + \sqrt{49} = 14 + 7 = 21$.
The correct way: the root of a sum is not the sum of the roots. Add first, then take the root: $\sqrt{196 + 49} = \sqrt{245} = 7\sqrt{5} \approx 15.652$.
Conclusion
The square root of 245 is irrational, equal to $7\sqrt{5} \approx 15.652$.
245 factors as $5 \times 7^2$, so the perfect square 49 pulls out as 7 and leaves √5 inside.
The value sits between 15 and 16 because 245 lies between the squares 225 and 256.
√245, √125, and √500 all share the same √5 core, which links them in simplified form.
To keep building radical skills with a teacher, explore Bhanzu's algebra tutor or get help with algebra.
Read More
Square root of 340 — another radical that simplifies, to 2√85
Square root of 5 — the √5 core inside √245
Square root of 25 — the perfect square that anchors nearby values
Squares and square roots — the rules behind simplifying radicals
What is a square root — the definition and notation in depth
Cube root of 100 — the same kind of question one index higher
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