What Is A Square Root?
The square root of a number $n$ is a value $r$ such that $r^2 = n$, the number that, multiplied by itself, gives $n$. The square root of 392 is the number whose square is 392.
No whole number does this: $19^2 = 361$ (too small) and $20^2 = 400$ (too big). So $\sqrt{392}$ lies between 19 and 20, close to 20, at about $19.799$.
Because 392 is not a perfect square, $\sqrt{392}$ is an irrational number - but it is not in its tidiest shape yet, because 392 hides a large perfect-square factor.
Where √392 Appears In Real Math
$\sqrt{392}$ is the diagonal of a $14 \times 14$ square — by the Pythagorean theorem the diagonal is $\sqrt{14^2 + 14^2} = \sqrt{196 + 196} = \sqrt{392} = 14\sqrt{2}$. Any square's diagonal is its side times $\sqrt{2}$, so a 14-unit square hands you $14\sqrt{2}$ directly, and that same $\sqrt{2}$ scaling is why a square photo enlarged to fill a diagonal frame grows by about 41 percent.
Quick Reference Table
Number $n$ | $\sqrt{n}$ (approx.) | Simplest radical form |
|---|---|---|
98 | 9.899 | $7\sqrt{2}$ |
128 | 11.314 | $8\sqrt{2}$ |
196 | 14 | $14$ (perfect square) |
200 | 14.142 | $10\sqrt{2}$ |
392 | 19.799 | $\mathbf{14\sqrt{2}}$ |
400 | 20 | $20$ (perfect square) |
450 | 21.213 | $15\sqrt{2}$ |
512 | 22.627 | $16\sqrt{2}$ |
Is The Square Root Of 392 Rational Or Irrational?
$\sqrt{392}$ is irrational - it cannot be written as a fraction $\frac{p}{q}$ of two integers, and its decimal neither ends nor repeats.
Quick reasoning. A whole number has a rational square root only when it is a perfect square such as 361 or 400. Since 392 sits between $19^2 = 361$ and $20^2 = 400$, it is not a perfect square, so $\sqrt{392}$ is irrational.
Is 392 a perfect square? No. Its prime factorisation is $2^3 \times 7^2$, and the prime 2 appears an odd number of times, so no integer squares to 392. Simplifying pulls out the paired factors and leaves the lone 2 behind:
$$\sqrt{392} = \sqrt{196 \times 2} = 14\sqrt{2}$$
The leftover $\sqrt{2}$ is famously irrational, which is what makes the whole value irrational. Keep $14\sqrt{2}$ in algebra; round to a decimal only when a numeric answer is required.
How Do You Find √392? (Prime Factorisation And Long Division)
Method 1: Prime factorisation (best for the exact form).
Break 392 into primes.
$$392 = 2 \times 196$$
$$392 = 2 \times 2 \times 98$$
$$392 = 2^3 \times 7^2$$
Group the factors into pairs; each pair leaves the radical as a single number.
$$\sqrt{392} = \sqrt{2^2 \times 7^2 \times 2}$$
$$\sqrt{392} = 2 \times 7 \times \sqrt{2}$$
$$\sqrt{392} = 14\sqrt{2}$$
Final answer: $\sqrt{392} = 14\sqrt{2}$.
Method 2: Long division (best for the decimal).
Pair the digits from the decimal point: $3,\overline{92}.\overline{00},\overline{00}$
Step 1: The largest square $\le 3$ is $1^2 = 1$; write 1, remainder $3 - 1 = 2$, bring down 92 to make 292.
Step 2: Double the quotient to get 2; find $d$ with $(20 + d)\times d \le 292$; $d = 9$ gives $29 \times 9 = 261$, remainder $31$. Quotient is 19.
Step 3: Bring down 00 to make 3100; double 19 to get 38; $(380 + d)\times d \le 3100$; $d = 7$ gives $387 \times 7 = 2709$, remainder 391. Quotient is 19.7.
Step 4: Bring down 00 to make 39100; double 197 to get 394; $(3940 + d)\times d \le 39100$; $d = 9$ gives $3949 \times 9 = 35541$. Quotient is 19.79.
Continuing gives $\sqrt{392} \approx 19.799$. The long division never terminates, which is the mark of an irrational value.
Examples Of Square Root Of 392
Example 1
Write $\sqrt{392}$ in simplest radical form.
Find the largest perfect-square factor of 392.
$$392 = 196 \times 2$$
$$\sqrt{392} = \sqrt{196} \times \sqrt{2}$$
$$\sqrt{392} = 14\sqrt{2}$$
The first instinct is to grab the first factor you recognise, often 4, and simplify in slow rounds; pulling out the largest perfect square, 196, in one move gets you straight to $14\sqrt{2}$.
Final answer: $14\sqrt{2}$.
Example 2
A common slip worth walking through: simplify $\sqrt{392}$ by splitting the sum.
Since $392 = 196 + 196$, the tempting move is to write $\sqrt{392} = \sqrt{196} + \sqrt{196} = 14 + 14 = 28$.
Check it.
$$28^2 = 784$$
But $784 \ne 392$, so 28 is far too big - a square root does not distribute over addition. Roots split over multiplication, not sums.
Use the product instead: $392 = 196 \times 2$, so $\sqrt{392} = 14\sqrt{2} \approx 19.8$.
Final answer: $\sqrt{392} = 14\sqrt{2}$, not 28.
Example 3
Evaluate $(\sqrt{392})^2$.
Squaring undoes the square root.
$$(\sqrt{392})^2 = 392$$
Final answer: $392$.
Example 4
Simplify $\sqrt{392} + \sqrt{50}$.
Simplify each radical to reveal like terms.
$$\sqrt{392} = 14\sqrt{2}$$
$$\sqrt{50} = 5\sqrt{2}$$
$$14\sqrt{2} + 5\sqrt{2} = 19\sqrt{2}$$
Radicals add only when the part under the root matches, exactly like combining $14x + 5x = 19x$.
Final answer: $19\sqrt{2} \approx 26.87$.
Common Mistakes
Mistake 1: Splitting the root over addition
Where it slips in: Whenever the number under the root can be written as a sum, like $392 = 196 + 196$.
Don't do this: Writing $\sqrt{392} = \sqrt{196} + \sqrt{196} = 28$.
The correct way: Square roots split over multiplication, never over addition. Factor into a product, $392 = 196 \times 2$, then $\sqrt{392} = 14\sqrt{2}$. The habit of factoring into a product first is what keeps the sign and the size honest, and it is the confusion that trips up more students on this problem than any arithmetic error.
Mistake 2: Not pulling out the largest perfect square
Where it slips in: Simplifying in small steps and stopping early.
Don't do this: Writing $\sqrt{392} = 2\sqrt{98}$ and calling it finished.
The correct way: $98$ still holds the perfect square 49, so $2\sqrt{98} = 2 \times 7\sqrt{2} = 14\sqrt{2}$. Always factor out the largest perfect square in one move to land in simplest form immediately.
Mistake 3: Rounding too early
Where it slips in: Multi-step problems where $\sqrt{392}$ appears mid-calculation.
Don't do this: Replace $\sqrt{392}$ with $19.799$ at the start and carry that through every step.
The correct way: Keep $14\sqrt{2}$ symbolically until the final line, then round once. The exact form is both more accurate and, in a sum like $14\sqrt{2} + 5\sqrt{2}$, far easier to combine.
Conclusion
The square root of 392 is $14\sqrt{2} \approx 19.799$: irrational, with 196 as its largest perfect-square factor. Factor into a product first, pull out the biggest square in one move, and keep $14\sqrt{2}$ until the final answer. To practise simplifying radicals with a teacher, explore Bhanzu's algebra tutor, a high school math tutor, or live math classes online. Want the method taught step by step? Book a free demo class.
Read More
Square Root of 396 — the very next number, simplified to 6√11
Square Root of 50 — a smaller "× root 2" radical, 5√2
Square Root of 8 — the simplest member of the root-2 family, 2√2
Square Root of 245 — the same method with a leftover root 5
Square Root 1 to 30 — the full reference table of roots in range
Square Root Tricks — fast estimation methods for roots without a calculator
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