What Is A Square Root?
The square root of a number $n$ is the value $r$ that satisfies $r \times r = n$. For 850, we want the number that, multiplied by itself, returns 850.
No whole number does this, because $29^2 = 841$ (too small) and $30^2 = 900$ (too big). So $\sqrt{850}$ lives between 29 and 30, much closer to 29.
The symbol $\sqrt{850}$ always means the positive (principal) root. The equation $x^2 = 850$ has two solutions, $x = \pm\sqrt{850}$, but the radical on its own points only to the positive one.
Where Does √850 Appear?
$\sqrt{850}$ shows up whenever a square has an area of 850 square units: its side length is exactly $5\sqrt{34}$, about $29.15$ units. It also appears as the distance between two points on a coordinate grid - the points $(0, 0)$ and $(15, 25)$ are $\sqrt{15^2 + 25^2} = \sqrt{225 + 625} = \sqrt{850}$ apart, straight from the distance formula. Any time a Pythagorean setup produces $850$ under the root, this same $5\sqrt{34}$ is the tidy answer.
Quick Reference Table
Number $n$ | $\sqrt{n}$ simplified | $\sqrt{n}$ (approx.) |
|---|---|---|
800 | $20\sqrt{2}$ | 28.2843 |
841 | 29 | 29.0000 |
850 | $\mathbf{5\sqrt{34}}$ | 29.1548 |
864 | $12\sqrt{6}$ | 29.3939 |
900 | 30 | 30.0000 |
8.5 | $\dfrac{\sqrt{34}}{2}$ | 2.9155 |
34 | $\sqrt{34}$ | 5.8310 |
3400 | $10\sqrt{34}$ | 58.3095 |
How Do You Simplify The Square Root Of 850?
Simplifying a radical means pulling out any perfect-square factor hiding inside it. Start by breaking 850 into primes:
$$850 = 2 \times 425$$ $$425 = 5 \times 85$$ $$85 = 5 \times 17$$ $$850 = 2 \times 5^2 \times 17$$
The only squared prime is $5^2$. A pair of 5s comes out of the radical as a single 5, and the leftover factors stay inside:
$$\sqrt{850} = \sqrt{5^2 \times 34}$$ $$\sqrt{850} = 5\sqrt{34}$$
Since $34 = 2 \times 17$ has no square factor left, $5\sqrt{34}$ is the simplest radical form. This is the same idea used across simplifying radical expressions.
Is The Square Root Of 850 Rational Or Irrational?
$\sqrt{850}$ is irrational - it cannot be written as a fraction of two integers, and its decimal never terminates or repeats.
Is 850 a perfect square? No. A whole number has a rational square root only when it is a perfect square (1, 4, 9, 16, 25, 36, ...). Because 850 falls between $29^2$ and $30^2$, it is not one, so its root is irrational.
You can also see it from the simplified form: $5\sqrt{34}$ contains $\sqrt{34}$, and 34 is not a perfect square either. Any exact decimal you write for $\sqrt{850}$ is only an approximation; the true value lives in the symbol $5\sqrt{34}$.
How Do You Find √850? (Long Division And Prime Factorization)
Prime factorization gives the exact simplified form, shown above: $\sqrt{850} = 5\sqrt{34}$. For the decimal, use the long division method, which handles any number, perfect square or not — the standard by-hand square-root algorithm.
Pair the digits outward from the decimal point:
$$8\ \overline{50}\ .\ \overline{00}\ \overline{00}\ \overline{00}$$
Find the largest square not exceeding the first group (8):
$$2^2 = 4 \le 8$$
Write 2 in the quotient, subtract, and bring down the next pair:
$$8 - 4 = 4$$ $$\text{bring down } 50 \text{ to get } 450$$
Double the quotient (2) to get 4, then find a digit $d$ with $(40 + d)\times d \le 450$:
$$49 \times 9 = 441 \le 450$$
Quotient is now 29; subtract and bring down the next pair:
$$450 - 441 = 9$$ $$\text{bring down } 00 \text{ to get } 900$$
Double 29 to get 58, then find $d$ with $(580 + d)\times d \le 900$:
$$581 \times 1 = 581 \le 900$$
Quotient becomes 29.1. Continue the same loop:
$$5825 \times 5 = 29125 \quad (\text{gives } 29.15)$$ $$58304 \times 4 = 233216 \quad (\text{gives } 29.154)$$
So $\sqrt{850} \approx 29.1548$. The process never ends, which is exactly what "irrational" means. If you already know $\sqrt{34} \approx 5.8310$, the fastest route is $5 \times 5.8310 = 29.155$.
Examples Of √850
Example 1
Simplify $\sqrt{850}$ using prime factorization.
$$850 = 2 \times 5^2 \times 17$$ $$\sqrt{850} = \sqrt{5^2}\times\sqrt{34}$$ $$\sqrt{850} = 5\sqrt{34}$$
Final answer: $5\sqrt{34}$.
Example 2
Where the simplification usually goes wrong. A common first move is to notice $850 = 25 \times 34$ and pull the 25 straight out, writing $\sqrt{850} = 25\sqrt{34}$. Does that hold? Check it: $25 \times 5.831 \approx 145.8$. But $\sqrt{850}$ has to sit between 29 and 30, so 145.8 is far too large. The slip is treating $\sqrt{25}$ as 25. The radical of a perfect square is its root, not the number itself:
$$\sqrt{25} = 5$$ $$\sqrt{850} = 5\sqrt{34} \approx 29.15$$
Final answer: $5\sqrt{34}$, not $25\sqrt{34}$.
Example 3
Estimate $\sqrt{850}$ between two perfect squares.
$$29^2 = 841$$ $$30^2 = 900$$ $$841 < 850 < 900$$
So $29 < \sqrt{850} < 30$. Since 850 is only 9 above 841, the root is just above 29.
Final answer: about $29.15$.
Example 4
Find the decimal from the simplified form. Using $\sqrt{34} \approx 5.8310$:
$$\sqrt{850} = 5\sqrt{34}$$ $$\sqrt{850} \approx 5 \times 5.8310$$ $$\sqrt{850} \approx 29.155$$
Final answer: $29.155$ (rounds to $29.1548$ at four places).
Example 5
A square field has an area of 850 square metres. How long is each side?
$$\text{side} = \sqrt{\text{area}} = \sqrt{850}$$ $$\text{side} = 5\sqrt{34} \text{ m}$$ $$\text{side} \approx 29.15 \text{ m}$$
Final answer: each side is $5\sqrt{34}$ m, about $29.15$ m.
Common Mistakes
Mistake 1: Pulling the perfect square out as itself
Where it slips in: Rewriting $850$ as $25 \times 34$ and moving 25 outside the radical.
Don't do this: Writing $\sqrt{850} = 25\sqrt{34}$.
The correct way: Take the square root of the perfect-square factor. $\sqrt{25} = 5$, so $\sqrt{850} = 5\sqrt{34}$. The habit that fixes this is a quick size check — the answer must land between 29 and 30, and $25\sqrt{34}$ never could.
Mistake 2: Stopping before the radical is fully simplified
Where it slips in: Factoring only partway, for example $850 = 2 \times 425$.
Don't do this: Writing $\sqrt{850} = \sqrt{2}\times\sqrt{425}$ and calling it done, since 425 still hides a $5^2$.
The correct way: Factor all the way to primes ($2 \times 5^2 \times 17$) so every perfect-square factor is caught. Only then is $5\sqrt{34}$ guaranteed simplest.
Mistake 3: Rounding too early
Where it slips in: Multi-step problems where $\sqrt{850}$ appears mid-calculation.
Don't do this: Replace $\sqrt{850}$ with $29.15$ at the start and carry that rounded value through every step.
The correct way: Keep $5\sqrt{34}$ symbolically until the final line. Early rounding compounds error across each multiplication and division.
Conclusion
The square root of 850 is $5\sqrt{34}$ exactly, about $29.1548$ as a decimal, and irrational because 850 is not a perfect square. Prime factorization gives the clean radical form, and long division delivers the decimal to any precision you need.
To go further with radicals and simplification alongside a teacher, explore Bhanzu's algebra tutor, our high school math tutor sessions, or math classes online. Want to work a set of these live? Book a free demo class.
Read More
Square root of 50 — another radical that simplifies to $5\sqrt{2}$.
Square root of 20 — a smaller worked simplification, $2\sqrt{5}$.
Squares and square roots — the full relationship, from perfect squares to radicals.
Perfect squares — the list you check a number against before simplifying.
Square root 1 to 30 — a reference table of exact and decimal roots.
Square root tricks — faster ways to estimate roots by hand.
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