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
Number $n$ | $\sqrt{n}$ (approx.) | Simplified form | Rational or Irrational |
|---|---|---|---|
225 | 15 | $15$ | Rational |
240 | 15.4919 | $\mathbf{4\sqrt{15}}$ | Irrational |
243 | 15.5885 | $9\sqrt{3}$ | Irrational |
245 | 15.6525 | $7\sqrt{5}$ | Irrational |
250 | 15.8114 | $5\sqrt{10}$ | Irrational |
256 | 16 | $16$ | Rational |
260 | 16.1245 | $2\sqrt{65}$ | Irrational |
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 $4\sqrt{15}$.
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