What Is A Square Root?
A square root of a number $n$ is a value $r$ with $r^2 = n$. The square root of 306 is the positive number that, multiplied by itself, gives 306.
The two nearest perfect squares set its range:
$$17^2 = 289$$
$$18^2 = 324$$
Since 306 lies between them, $\sqrt{306}$ lies between 17 and 18. This pairing of a number with its root is the heart of squares and square roots, and it tells us the decimal starts with 17.
Where Does √306 Appear?
$\sqrt{306}$ is the exact diagonal of a rectangle with sides 15 and 9, because the Pythagorean theorem gives $\sqrt{15^2 + 9^2} = \sqrt{225 + 81} = \sqrt{306}$. It is also the side length of a square whose area is 306 square units, close to 17.49 units. Whenever a factor pair of $9 \times 34$ turns up under a root, the answer folds down to $3\sqrt{34}$, which is far tidier to carry through a calculation than the raw decimal.
Quick Reference Table
Number $n$ | $\sqrt{n}$ (approx.) | Simplified | Rational or Irrational |
|---|---|---|---|
289 | 17 | $17$ | Rational |
300 | 17.3205 | $10\sqrt{3}$ | Irrational |
306 | 17.4929 | $3\sqrt{34}$ | Irrational |
320 | 17.8885 | $8\sqrt{5}$ | Irrational |
324 | 18 | $18$ | Rational |
For the smaller roots and their patterns, see the reference on square roots from 1 to 30.
Is The Square Root Of 306 Rational Or Irrational?
$\sqrt{306}$ is irrational. After simplifying to $3\sqrt{34}$, the remaining $\sqrt{34}$ is irrational, and tripling an irrational number keeps it irrational.
The quick reason. A whole number has a rational root only when it is a perfect square. 306 is not, and its simplified core 34 is not either. By the standard account of an irrational number, such a value has a decimal that never terminates and never repeats.
Why 34 stays inside. $34 = 2 \times 17$, two distinct primes with no pair between them, so no further perfect square can be extracted. The exact value is $3\sqrt{34}$; anything like 17.4929 is only a rounded approximation.
How Do You Find √306? (Prime Factorization And Long Division)
Method 1: Prime factorization (for the simplified form)
Step 1: Break 306 into primes.
$$306 = 2 \times 153$$
$$153 = 3 \times 51 = 3 \times 3 \times 17$$
$$306 = 2 \times 3^2 \times 17$$
Step 2: The pair $3^2$ leaves the radical as 3. The remaining factors, $2 \times 17 = 34$, stay inside.
$$\sqrt{306} = \sqrt{3^2 \times 34} = 3\sqrt{34}$$
Step 3: 34 has no repeated prime factor, so nothing else comes out.
$$\sqrt{306} = 3\sqrt{34}$$
The pairing step is the same one used across simplifying radical expressions.
Method 2: Long division (for the decimal)
Step 1: Pair the digits: $3,06$, then zeros after the point.
Step 2: The largest square at most 3 is $1^2 = 1$. First digit is 1.
$$3 - 1 = 2$$
Step 3: Bring down 06 to get 206. Double 1 to get 2; find $d$ with $(20 + d) \times d \leq 206$. Testing $d = 7$: $27 \times 7 = 189$. Quotient 17.
$$206 - 189 = 17$$
Step 4: Bring down zeros to get 1700. Double 17 to get 34; find $d$ with $(340 + d) \times d \leq 1700$. Testing $d = 4$: $344 \times 4 = 1376$. Quotient 17.4.
$$1700 - 1376 = 324$$
Step 5: Continue the rounds to reach 17.4929.
$$\sqrt{306} \approx 17.4929$$
A quick estimate confirms it: $3\sqrt{34}$, and since $\sqrt{34} \approx 5.83$, the product is about 17.49. The square root tricks reference collects more of these estimation shortcuts.
Examples Of √306
Example 1
Simplify $\sqrt{306}$.
Factor: $306 = 3^2 \times 34$.
Pull the pair out: $\sqrt{3^2 \times 34} = 3\sqrt{34}$.
Final answer: $3\sqrt{34}$.
Example 2
The tempting shortcut that does not work: simplifying $\sqrt{306}$.
A frequent attempt splits it as $\sqrt{306} = \sqrt{2} \times \sqrt{153}$ and stops, calling it simplified.
Wrong path: 153 still holds a perfect square, since $153 = 9 \times 17$.
Where it breaks: leaving 153 inside misses the pair of 3s hiding in it.
The rescue: factor fully to $2 \times 3^2 \times 17$, so the pair $3^2$ is visible and comes out as 3.
Final answer: $3\sqrt{34}$.
Example 3
Estimate $\sqrt{306}$ to the nearest whole number.
The nearest perfect squares are $17^2 = 289$ and $18^2 = 324$.
306 is 17 above 289 and 18 below 324, so it sits near the middle.
Since $\sqrt{306} \approx 17.49$ is just below the midpoint, the nearest whole number is 17.
Final answer: 17.
Example 4
Solve $x^2 = 306$.
Take the square root of both sides.
$$x = \pm\sqrt{306}$$
$$x = \pm 3\sqrt{34} \approx \pm 17.4929$$
Final answer: $x = 3\sqrt{34}$ or $x = -3\sqrt{34}$.
Example 5
A rectangle has sides 15 and 9. Find its diagonal.
The diagonal is the hypotenuse of a right triangle with legs 15 and 9.
$$d = \sqrt{15^2 + 9^2}$$
$$d = \sqrt{225 + 81} = \sqrt{306} = 3\sqrt{34}$$
Final answer: $3\sqrt{34} \approx 17.49$ units.
Common Mistakes
Mistake 1: Not factoring all the way down
Where it slips in: the simplification step. Students first meeting 306 often stop at $\sqrt{2} \times \sqrt{153}$ and miss the perfect square hiding inside 153.
Don't do this: leaving a factor like 153 or 34 uninspected for pairs.
The correct way: break every factor into primes until no more pairs appear. Then $306 = 2 \times 3^2 \times 17$ gives $3\sqrt{34}$.
Mistake 2: Multiplying the outside and inside numbers together
Where it slips in: reading $3\sqrt{34}$ as one product.
Don't do this: writing $3\sqrt{34} = \sqrt{102}$ by pulling the 3 back under the radical incorrectly.
The correct way: to move 3 inside, square it first: $3\sqrt{34} = \sqrt{9 \times 34} = \sqrt{306}$, not $\sqrt{102}$.
Mistake 3: Rounding $\sqrt{34}$ too early
Where it slips in: multi-step problems using $3\sqrt{34}$.
Don't do this: replacing $\sqrt{34}$ with 5.83 at the start and carrying that rounded value through.
The correct way: keep $3\sqrt{34}$ exact until the final step, then round once.
Conclusion
The square root of 306 simplifies to $3\sqrt{34}$ and rounds to about 17.4929. The key step is factoring 306 completely, spotting the pair of 3s, and pulling out a single 3 while 34 stays inside. Practise that full-factoring habit and radical simplification stops feeling like guesswork. To keep going with a teacher, try a Bhanzu algebra tutor, get one-on-one help with algebra, or join live math classes online. You can also book a free demo class.
Read More
Square Root of 260 — a nearby root that simplifies to $2\sqrt{65}$.
Square Root of 340 — another radical worked by the same pairing method.
Square Root of 255 — a root with no perfect-square factor to remove.
Square Root 1 to 25 — a lookup table of the smaller roots.
Exponents — how a radical relates to a fractional power.
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