Square Root of 306 - Value, Simplified Form & Steps

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

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.