Square Root of 245 — Value, Simplification, Steps

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.

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