Where Does √425 Appear?
$\sqrt{425}$ is the exact length of the diagonal of a rectangle with sides 20 and 5, since the Pythagorean theorem gives $\sqrt{20^2 + 5^2} = \sqrt{400 + 25} = \sqrt{425}$. It is also the side of a square whose area is 425 square units, roughly 20.62 units long. Any time a computation produces $25 \times 17$ under a root, the answer collapses to $5\sqrt{17}$, so recognising the hidden square saves a messy decimal.
What Is A Square Root?
A square root of a number $n$ is a value $r$ with $r^2 = n$. The square root of 425 is the positive number that, squared, returns 425.
The two closest perfect squares fix its size:
$$20^2 = 400$$
$$21^2 = 441$$
So $\sqrt{425}$ sits between 20 and 21. The link between a number and its root is the core of squares and square roots, and here it tells us the decimal begins with 20.
Quick Reference Table
Number $n$ | $\sqrt{n}$ (approx.) | Simplified | Rational or Irrational |
|---|---|---|---|
400 | 20 | $20$ | Rational |
405 | 20.1246 | $9\sqrt{5}$ | Irrational |
420 | 20.4939 | $2\sqrt{105}$ | Irrational |
425 | 20.6155 | $5\sqrt{17}$ | Irrational |
441 | 21 | $21$ | Rational |
450 | 21.2132 | $15\sqrt{2}$ | Irrational |
The wider pattern of perfect squares and their neighbours is laid out in square roots from 1 to 30.
Is The Square Root Of 425 Rational Or Irrational?
$\sqrt{425}$ is irrational. Even after simplifying to $5\sqrt{17}$, the leftover $\sqrt{17}$ is irrational, and multiplying an irrational number by 5 keeps it irrational.
The quick reason. A whole number has a rational root only if it is a perfect square. 425 is not, and its simplified core, 17, is prime. According to the standard definition of a square root, a non-perfect-square integer always yields an irrational value.
What simplifying does and does not do. Writing $\sqrt{425}$ as $5\sqrt{17}$ makes the number cleaner to handle, but it does not make it rational. The exact value lives in $5\sqrt{17}$; any decimal such as 20.6155 is a rounded stand-in.
How Do You Find √425? (Prime Factorization And Long Division)
Method 1: Prime factorization (for the simplified form)
Step 1: Break 425 into primes.
$$425 = 5 \times 85$$
$$85 = 5 \times 17$$
$$425 = 5^2 \times 17$$
Step 2: Take the square root of each part. The pair $5^2$ leaves the radical as 5.
$$\sqrt{425} = \sqrt{5^2 \times 17} = 5\sqrt{17}$$
Step 3: 17 is prime, so nothing else comes out.
$$\sqrt{425} = 5\sqrt{17}$$
This is the exact answer. The same pairing idea drives every problem in simplifying radical expressions.
Method 2: Long division (for the decimal)
Step 1: Pair the digits: $4,25$, then zeros after the point.
Step 2: The largest square at most 4 is $2^2 = 4$. First digit is 2 (the tens place).
$$4 - 4 = 0$$
Step 3: Bring down 25. Double 2 to get 4; find $d$ with $(40 + d) \times d \leq 25$. Here $d = 0$. Quotient so far is 20.
Step 4: Bring down zeros to get 2500. Double 20 to get 40; find $d$ with $(400 + d) \times d \leq 2500$. Testing $d = 6$: $406 \times 6 = 2436$. Quotient 20.6.
$$2500 - 2436 = 64$$
Step 5: Continue the rounds to reach 20.6155.
$$\sqrt{425} \approx 20.6155$$
Estimation offers a fast check: $5 \times \sqrt{17}$, and since $\sqrt{17} \approx 4.123$, the product is about 20.6. For more of these shortcuts, see the square root tricks reference.
Examples Of √425
Example 1
Simplify $\sqrt{425}$.
Factor: $425 = 5^2 \times 17$.
Pull the pair out: $\sqrt{5^2 \times 17} = 5\sqrt{17}$.
Final answer: $5\sqrt{17}$.
Example 2
Where students lose the mark: simplifying $\sqrt{425}$.
A common attempt stops halfway and writes $\sqrt{425} = 25\sqrt{17}$, reasoning "25 is the perfect square, so 25 comes out."
Wrong path: that treats $\sqrt{25}$ as 25, but $\sqrt{25} = 5$, not 25.
Where it breaks: only the square root of the perfect square leaves the radical, not the perfect square itself.
The rescue: $\sqrt{5^2} = 5$, so the answer is $5\sqrt{17}$.
Final answer: $5\sqrt{17}$, not $25\sqrt{17}$.
Example 3
Find the decimal value of $\sqrt{425}$ to two places.
Use the simplified form: $5\sqrt{17}$.
$$\sqrt{17} \approx 4.1231$$
$$5 \times 4.1231 = 20.6155$$
Final answer: $\approx 20.62$.
Example 4
Solve $x^2 = 425$.
Take the square root of both sides.
$$x = \pm\sqrt{425}$$
$$x = \pm 5\sqrt{17} \approx \pm 20.6155$$
Final answer: $x = 5\sqrt{17}$ or $x = -5\sqrt{17}$.
Example 5
A rectangle has sides 20 and 5. Find its diagonal.
The diagonal is the hypotenuse of a right triangle with legs 20 and 5.
$$d = \sqrt{20^2 + 5^2}$$
$$d = \sqrt{400 + 25} = \sqrt{425} = 5\sqrt{17}$$
Final answer: $5\sqrt{17} \approx 20.62$ units.
Common Mistakes
Mistake 1: Bringing the whole square out instead of its root
Where it slips in: the simplification step. Students first meeting $425 = 25 \times 17$ often write 25 outside the radical.
Don't do this: $\sqrt{425} = 25\sqrt{17}$.
The correct way: $\sqrt{25} = 5$, so only 5 leaves the radical. The answer is $5\sqrt{17}$.
Mistake 2: Stopping at the wrong factor
Where it slips in: picking $\sqrt{425} = \sqrt{5} \times \sqrt{85}$ and calling it simplified.
Don't do this: leaving 85 inside, since $85 = 5 \times 17$ still holds a factor of 5.
The correct way: factor fully to $5^2 \times 17$ so the perfect square is obvious, giving $5\sqrt{17}$.
Mistake 3: Rounding $\sqrt{17}$ too early
Where it slips in: multi-step problems using $5\sqrt{17}$.
Don't do this: replacing $\sqrt{17}$ with 4.12 at the start and multiplying everything by that.
The correct way: keep $5\sqrt{17}$ exact until the last step, then round once to the precision you need.
Conclusion
The square root of 425 simplifies to $5\sqrt{17}$ and rounds to about 20.6155. The move that matters is spotting the hidden perfect square 25, pulling out its root of 5, and leaving the prime 17 inside. That single habit unlocks most radical simplifications you will meet at this level. To go further with a teacher, work with a Bhanzu algebra tutor, pair up with a high school math tutor for exam-level practice, or explore ongoing math tutoring. You can also book a free demo class.
Read More
Square Root of 441 — the perfect square just above 425, equal to 21.
Square Root of 340 — a nearby non-perfect-square root worked the same way.
Square Root of 384 — another radical that simplifies, to $8\sqrt{6}$.
Square Root 1 to 25 — a lookup table of the smaller roots.
Rational Exponents — writing $\sqrt{425}$ as $425^{1/2}$.
Was this article helpful?
Your feedback helps us write better content