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 272 is the number that, multiplied by itself, gives 272.
No integer does this, because $16^2 = 256$ (too small) and $17^2 = 289$ (too big). So $\sqrt{272}$ lies between 16 and 17, close to 16.5.
The number under the radical sign - the radicand - is 272. Simplifying means rewriting it as a perfect square times a leftover, then taking the root of the perfect square.
Where Does √272 Appear?
$\sqrt{272}$ is the diagonal of a $4 \times 16$ rectangle - the Pythagorean theorem gives $\sqrt{4^2 + 16^2} = \sqrt{16 + 256} = \sqrt{272}$. Since $\sqrt{272} = 4\sqrt{17}$, it is also exactly 4 times the diagonal of a $1 \times 4$ rectangle, whose diagonal is $\sqrt{17}$. Written as a rational exponent, the same value is $272^{1/2}$, the form that shows up when radicals meet exponent rules.
Quick Reference Table
Number $n$ | $\sqrt{n}$ (approx.) | Simplified form | Rational or Irrational |
|---|---|---|---|
256 | 16 | $16$ | Rational |
260 | 16.1245 | $2\sqrt{65}$ | Irrational |
269 | 16.4012 | $\sqrt{269}$ | Irrational |
272 | 16.4924 | $\mathbf{4\sqrt{17}}$ | Irrational |
275 | 16.5831 | $5\sqrt{11}$ | Irrational |
288 | 16.9706 | $12\sqrt{2}$ | Irrational |
289 | 17 | $17$ | Rational |
Is The Square Root Of 272 Rational Or Irrational?
$\sqrt{272}$ 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. 272 is not a perfect square, so $\sqrt{272}$ is irrational.
The prime-factor reason. Factor the radicand:
$$272 = 2^4 \times 17$$
For a square root to be rational, every prime must appear an even number of times. Here 17 appears once - an odd power - so the root cannot resolve to a whole number or a fraction. That lone 17 stays trapped inside $\sqrt{17}$ after simplification.
How Do You Find √272?
Prime factorization gives the exact simplified form; long division gives the decimal.
Prime Factorization (Exact Form)
Break 272 into primes and pair off the squares:
$$272 = 2^4 \times 17$$
$$\sqrt{272} = \sqrt{2^4 \times 17}$$
$$\sqrt{272} = \sqrt{2^4} \times \sqrt{17}$$
$$\sqrt{272} = 4\sqrt{17}$$
The largest perfect-square factor of 272 is $16 = 2^4$, which comes out as 4. What remains, $\sqrt{17}$, has no perfect-square factor, so $4\sqrt{17}$ is the simplest radical form.
Long Division (Decimal Value)
Step 1: Pair the digits from the decimal point: $\overline{2},\overline{72}.\overline{00},\overline{00}$.
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 72 to make 172. Double the quotient: $1 \times 2 = 2$. Find $d$ with $(20 + d),d \leq 172$: $d = 6$ gives $26 \times 6 = 156$. Quotient 16; remainder 16.
Step 4: Add the decimal point, bring down 00 to make 1600. Double 16 to get 32. Find $d$ with $(320 + d),d \leq 1600$: $d = 4$ gives $324 \times 4 = 1296$. Quotient 16.4; remainder 304.
Step 5: Bring down 00 to make 30400. Double 164 to get 328. Find $d$ with $(3280 + d),d \leq 30400$: $d = 9$ gives $3289 \times 9 = 29601$. Quotient 16.49; remainder 799.
Step 6: Bring down 00 to make 79900. Double 1649 to get 3298. Find $d$ with $(32980 + d),d \leq 79900$: $d = 2$ gives $32982 \times 2 = 65964$. Quotient 16.492; remainder 13936.
Continuing gives $\sqrt{272} \approx 16.492$, and to four decimals $\sqrt{272} \approx 16.4924$.
Examples Of √272
Example 1: Confirm the simplified form
Show that $4\sqrt{17}$ squares back to 272.
$$\left(4\sqrt{17}\right)^2 = 4^2 \times \left(\sqrt{17}\right)^2$$
$$= 16 \times 17$$
$$= 272$$
Final answer: $\left(4\sqrt{17}\right)^2 = 272$, so the simplification checks out.
Example 2: The mistake worth making once
Simplify $\sqrt{272}$.
The tempting path: A student writes $\sqrt{272} = \sqrt{4 \times 68} = 2\sqrt{68}$ and stops there.
Where it breaks: $2\sqrt{68}$ is correct but not simplest: 68 still hides a perfect square, since $68 = 4 \times 17$. Students meeting radical simplification often extract the first perfect square they notice and stop too soon.
The rescue: Keep factoring until nothing square remains:
$$2\sqrt{68} = 2\sqrt{4 \times 17} = 2 \times 2\sqrt{17} = 4\sqrt{17}$$
Final answer: $\sqrt{272} = 4\sqrt{17}$.
Example 3: Multiply two radicals
Simplify $\sqrt{272} \times \sqrt{17}$.
$$\sqrt{272} \times \sqrt{17} = \sqrt{272 \times 17}$$
$$= \sqrt{4624}$$
$$= 68$$
Final answer: $\sqrt{272} \times \sqrt{17} = 68$, because $4\sqrt{17} \times \sqrt{17} = 4 \times 17 = 68$.
Example 4: Estimate between perfect squares
Estimate $\sqrt{272}$ to one decimal place.
Since $16^2 = 256$ and $17^2 = 289$, the root lies between 16 and 17. The gap $272 - 256 = 16$ out of the interval width $289 - 256 = 33$ gives roughly $16 + \frac{16}{33} \approx 16.5$.
Final answer: $\sqrt{272} \approx 16.5$, close to the true $16.4924$.
Common Mistakes
Mistake 1: Stopping the simplification early
Where it slips in: Pulling out one perfect square and leaving another behind.
Don't do this: Writing $\sqrt{272} = 2\sqrt{68}$ as the final answer.
The correct way: Factor fully, so $272 = 16 \times 17$, which means the answer is $4\sqrt{17}$, where 17 has no square factor left.
Mistake 2: Calling 272 a perfect square
Where it slips in: Rushing the rational-or-irrational check.
Don't do this: Expecting $\sqrt{272}$ to be a whole number.
The correct way: 272 sits between $256 = 16^2$ and $289 = 17^2$, so its root is irrational.
Mistake 3: Rounding too early
Where it slips in: Replacing $\sqrt{272}$ with 16.49 at the start of a longer calculation.
Don't do this: Carrying a rounded 16.49 through every step.
The correct way: Keep the exact form $4\sqrt{17}$ until the final line, then round once.
Conclusion
The square root of 272 is $4\sqrt{17} \approx 16.4924$: not a perfect square, irrational, and simplified by extracting the perfect-square factor 16 to leave $\sqrt{17}$ under the radical. Prime factorization gives the exact form, and long division gives the decimal. To take radical work further with a teacher, explore Bhanzu's algebra tutor sessions, join focused algebra classes, or work with a high school math tutor.
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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 roots between perfect squares.
Squares and Square Roots — the core rules linking squaring and its inverse.
Square Root of 260 — a neighbouring non-perfect square that simplifies to $2\sqrt{65}$.
Square Root of 255 — another nearby root that stays fully under the radical.
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