Square Root of 240 - Value, Simplified Form, and Examples

What Is A Square Root?

The square root of a number $n$ is a value $r$ such that $r^2 = n$. The square root of 240 is the number that, multiplied by itself, gives 240.

No integer does this, because $15^2 = 225$ (too small) and $16^2 = 256$ (too big). So $\sqrt{240}$ lies between 15 and 16, closer to 15.5.

The radicand is the number under the radical sign - here, 240. Simplifying a square root means rewriting that radicand as a perfect square times a leftover factor, then taking the root of the perfect square.

Where Does √240 Appear?

A square whose area is $240$ square units has sides of length $\sqrt{240} = 4\sqrt{15}$ units - so any time an area of 240 needs to become a side length, this value is the answer. It also surfaces in the quadratic formula: an equation whose discriminant equals 240 produces roots that carry the irrational $\sqrt{240}$. And because $240 = 16 \times 15$, the number is a classic teaching case for pulling a perfect square out from under a radical.

Quick Reference Table

Is The Square Root Of 240 Rational Or Irrational?

$\sqrt{240}$ is irrational - it cannot be written as a fraction $\frac{p}{q}$ of two integers, and its decimal neither terminates nor repeats.

The quick test. A whole number has a rational square root only when it is a perfect square. 240 is not a perfect square, so $\sqrt{240}$ is irrational.

The prime-factor reason. Factor the radicand:

$$240 = 2^4 \times 3 \times 5$$

For a square root to be rational, every prime must appear an even number of times. Here 3 and 5 each appear once - odd powers - so the root cannot resolve to a whole number or a fraction. The same fact leaves $\sqrt{15}$ trapped under the radical after simplification.

How Do You Find √240?

Two methods matter: prime factorization gives the exact simplified form, and long division gives the decimal.

Prime Factorization (Exact Form)

Break 240 into primes and pair them off:

$$240 = 2^4 \times 3 \times 5$$

$$\sqrt{240} = \sqrt{2^4 \times 3 \times 5}$$

$$\sqrt{240} = \sqrt{2^4} \times \sqrt{15}$$

$$\sqrt{240} = 4\sqrt{15}$$

The largest perfect-square factor of 240 is $16 = 2^4$, which comes out as 4. What remains, $\sqrt{15}$, has no perfect-square factor, so $4\sqrt{15}$ is the simplest radical form.

Long Division (Decimal Value)

Step 1: Pair the digits from the decimal point: $\overline{2},\overline{40}.\overline{00},\overline{00}$.

Step 2: The largest square $\leq 2$ is $1$ ($1^2 = 1$). First quotient digit is 1; remainder $2 - 1 = 1$.

Step 3: Bring down 40 to make 140. Double the quotient: $1 \times 2 = 2$. Find $d$ with $(20 + d),d \leq 140$: $d = 5$ gives $25 \times 5 = 125$. Quotient 15; remainder 15.

Step 4: Add the decimal point, bring down 00 to make 1500. Double 15 to get 30. Find $d$ with $(300 + d),d \leq 1500$: $d = 4$ gives $304 \times 4 = 1216$. Quotient 15.4; remainder 284.

Step 5: Bring down 00 to make 28400. Double 154 to get 308. Find $d$ with $(3080 + d),d \leq 28400$: $d = 9$ gives $3089 \times 9 = 27801$. Quotient 15.49; remainder 599.

Continuing one more place gives $\sqrt{240} \approx 15.491$, and to four decimals $\sqrt{240} \approx 15.4919$.

Examples Of √240

Example 1: Confirm the simplified form

Show that $4\sqrt{15}$ squares back to 240.

$$\left(4\sqrt{15}\right)^2 = 4^2 \times \left(\sqrt{15}\right)^2$$

$$= 16 \times 15$$

$$= 240$$

Final answer: $\left(4\sqrt{15}\right)^2 = 240$, so the simplification checks out.

Example 2: A common slip worth walking through

Simplify $\sqrt{240}$.

The tempting path: A student writes $\sqrt{240} = \sqrt{4 \times 60} = 2\sqrt{60}$ and stops.

Where it breaks: $2\sqrt{60}$ is correct but not simplest: 60 still contains a perfect square, since $60 = 4 \times 15$. Students meeting radical simplification for the first time usually pull out the first perfect square they spot and stop early.

The rescue: Keep factoring until nothing square is left:

$$2\sqrt{60} = 2\sqrt{4 \times 15} = 2 \times 2\sqrt{15} = 4\sqrt{15}$$

Final answer: $\sqrt{240} = 4\sqrt{15}$.

Example 3: Multiply two radicals

Simplify $\sqrt{240} \times \sqrt{15}$.

$$\sqrt{240} \times \sqrt{15} = \sqrt{240 \times 15}$$

$$= \sqrt{3600}$$

$$= 60$$

Final answer: $\sqrt{240} \times \sqrt{15} = 60$. The irrational parts cancel because $4\sqrt{15} \times \sqrt{15} = 4 \times 15 = 60$.

Example 4: Estimate without a calculator

Estimate $\sqrt{240}$ to one decimal place.

Since $15^2 = 225$ and $16^2 = 256$, the root is between 15 and 16. The gap $240 - 225 = 15$ out of the interval width $256 - 225 = 31$ gives roughly $15 + \frac{15}{31} \approx 15.5$.

Final answer: $\sqrt{240} \approx 15.5$, close to the true $15.4919$.

Common Mistakes

Mistake 1: Calling 240 a perfect square

Where it slips in: Rushing the rational-or-irrational check.

Don't do this: Assuming $\sqrt{240}$ resolves to a whole number because 240 is "round".

The correct way: Only perfect squares (225, 256, ...) give whole-number roots. 240 lies between them, so $\sqrt{240}$ is irrational.

Mistake 2: Stopping the simplification early

Where it slips in: Pulling out one perfect square and leaving another behind.

Don't do this: Writing $\sqrt{240} = 2\sqrt{60}$ as the final answer.

The correct way: Factor the radicand fully, so $240 = 16 \times 15$, which means the answer is $4\sqrt{15}$, where 15 has no square factor left.

Mistake 3: Rounding too early in a longer problem

Where it slips in: Replacing $\sqrt{240}$ with 15.49 at the start of a multi-step calculation.

Don't do this: Carrying a rounded 15.49 through every step.

The correct way: Keep the exact form $4\sqrt{15}$ until the final line, then round once. Early rounding compounds error.

Conclusion

The square root of 240 is $4\sqrt{15} \approx 15.4919$: not a perfect square, irrational, and simplified by pulling the perfect-square factor 16 out from under the radical. Prime factorization gives the exact form; long division gives the decimal. To take radical simplification further with a teacher, explore Bhanzu's algebra tutor sessions, get targeted help with algebra, or join structured math classes online.

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Read More

• Square Root 1 to 30 — every square root from 1 to 30 in one reference table.

• Simplifying Radical Expressions — the full method for reducing any radical to simplest form.

• Square Root Tricks — faster ways to estimate and compute roots by hand.

• Squares and Square Roots — the core rules linking squaring and its inverse.

• Square Root of 245 — a neighbouring non-perfect square that simplifies to $7\sqrt{5}$.

• Prove That Root 3 Is Irrational — the contradiction proof behind the $\sqrt{15}$ in $4\sqrt{15}$.