Square Root of 272 - Value, Simplified Form, and Steps

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

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.