What Is A Square Root?
The square root of a number $n$ is the value $r$ for which $r^2 = n$. The square root of 338 is the number that, multiplied by itself, returns 338.
No whole number does that exactly. $18^2 = 324$ is too small and $19^2 = 361$ is too big, so $\sqrt{338}$ lands between 18 and 19. The radical still simplifies, though, because 338 hides a perfect-square factor.
Where Does √338 Appear?
$\sqrt{338}$ is the diagonal of a square whose side is 13. In a square, the diagonal equals the side length times $\sqrt{2}$, so a side of 13 gives a diagonal of $13\sqrt{2} = \sqrt{338}$. The same value is the hypotenuse of a right isosceles triangle with both legs equal to 13. Any 45-45-90 layout built on a length of 13, a square tile, a corner brace, a diagonal cut, carries this exact measurement.
Quick Reference Table
Number $n$ | $\sqrt{n}$ (simplified) | $\sqrt{n}$ (approx.) |
|---|---|---|
288 | $12\sqrt{2}$ | 16.9706 |
324 | 18 | 18 |
338 | $\mathbf{13\sqrt{2}}$ | 18.3848 |
360 | $6\sqrt{10}$ | 18.9737 |
361 | 19 | 19 |
392 | $14\sqrt{2}$ | 19.7990 |
How Do You Find √338? (Prime Factorization)
The fastest route for 338 is prime factorization, because it exposes a square factor you can pull out. Work one step per line.
Step 1: Break 338 into prime factors. $338 = 2 \times 169$ $169 = 13 \times 13$ $338 = 2 \times 13^2$
Step 2: Split the radical over the factors. $\sqrt{338} = \sqrt{13^2 \times 2}$ $= \sqrt{13^2} \times \sqrt{2}$
Step 3: Take the perfect square outside. $\sqrt{13^2} = 13$ $\sqrt{338} = 13\sqrt{2}$
That is the simplest radical form. This is the whole idea behind simplifying radical expressions: find the largest perfect-square factor and move its root outside the sign.
For the decimal, multiply by the known value of $\sqrt{2}$. $\sqrt{2} \approx 1.41421$ $13 \times 1.41421 = 18.38473$ So $\sqrt{338} \approx 18.3848$.
You can confirm the same value by long division: $18^2 = 324 \leq 338$, first digit 18 $338 - 324 = 14$, bring down $00$ to get $1400$ $363 \times 3 = 1089 \leq 1400$, answer $18.3$ $1400 - 1089 = 311$, bring down $00$ to get $31100$ $3668 \times 8 = 29344 \leq 31100$, answer $18.38$ Both methods agree: $\sqrt{338} \approx 18.3848$.
Is The Square Root Of 338 Rational Or Irrational?
$\sqrt{338}$ is an irrational number. Even though it simplifies neatly to $13\sqrt{2}$, the leftover $\sqrt{2}$ is itself irrational, so the whole product is irrational.
Look at the factorization again. In $338 = 2 \times 13^2$, the prime 2 appears an odd number of times (once). For a whole number to be a perfect square, every prime in its factorization must appear an even number of times. The lone 2 breaks that condition, so 338 is not a perfect square and $\sqrt{338}$ cannot be a whole number or a fraction.
Examples Of √338
Example 1
Simplify $\sqrt{338}$ to simplest radical form.
$338 = 2 \times 13^2$ $\sqrt{338} = \sqrt{13^2} \times \sqrt{2}$ $= 13\sqrt{2}$ Final answer: $13\sqrt{2}$.
Example 2
A student writes $\sqrt{338} = \sqrt{338}$ and says it cannot be simplified because 338 is even but not a perfect square. Where is the slip?
The wrong path first. It is true 338 is not a perfect square, but "not a perfect square" does not mean "cannot be simplified." The real test is whether any perfect square divides 338. $338 = 2 \times 169$ $169 = 13^2$, a perfect square So pull it out. $\sqrt{338} = 13\sqrt{2}$ Final answer: $13\sqrt{2}$, not left as $\sqrt{338}$.
Example 3
Evaluate $(\sqrt{338})^2$.
Squaring reverses the square root. $(\sqrt{338})^2 = 338$ As a check, $(13\sqrt{2})^2 = 13^2 \times 2 = 169 \times 2 = 338$. Final answer: 338.
Example 4
Estimate $\sqrt{338}$ to the nearest whole number.
$18^2 = 324$ $19^2 = 361$ 338 is between 324 and 361, and closer to 324. Final answer: 18.
Example 5
Find $\dfrac{\sqrt{338}}{\sqrt{2}}$ without a calculator.
Combine under one radical. $\dfrac{\sqrt{338}}{\sqrt{2}} = \sqrt{\dfrac{338}{2}}$ $= \sqrt{169}$ $= 13$ Final answer: 13.
Common Mistakes
Mistake 1: Assuming a non-perfect-square cannot be simplified
Where it slips in: A learner sees that 338 is not a perfect square and stops.
Don't do this: Leaving the answer as $\sqrt{338}$.
The correct way: Check for a perfect-square factor first. Many students conflate "irrational" with "already simplified." Here $169 = 13^2$ divides 338, so $\sqrt{338} = 13\sqrt{2}$.
Mistake 2: Simplifying to the wrong coefficient
Where it slips in: Rushing the factor pair.
Don't do this: Writing $\sqrt{338} = 2\sqrt{169}$ or $\sqrt{338} = 26\sqrt{...}$.
The correct way: Move only the square root of the perfect square outside. Since $169 = 13^2$, its root is 13, leaving $\sqrt{2}$ inside: $13\sqrt{2}$.
Mistake 3: Squaring $13\sqrt{2}$ incorrectly
Where it slips in: Verifying the answer.
Don't do this: Writing $(13\sqrt{2})^2 = 13 \times 2 = 26$.
The correct way: Square both parts: $(13\sqrt{2})^2 = 13^2 \times (\sqrt{2})^2 = 169 \times 2 = 338$.
Conclusion
The square root of 338 is an irrational number that simplifies cleanly to $13\sqrt{2}$, about $18.3848$, because $338 = 2 \times 13^2$ hands you a perfect-square factor. Prime factorization does the real work; long division only confirms the decimal. To build fluency with radicals alongside a teacher, try Bhanzu's help with algebra resources or one-to-one math tutoring. You can also book a free demo class to see how the method is taught step by step.
Read More
Square root tricks for fast estimation and simplification
Square root 1 to 30 as a reference chart
Square root of 360, a nearby simplifiable root
Square root of 93, an irrational root that will not reduce
Square root of 365 for a similar value
Squares and square roots for the underlying concept
Was this article helpful?
Your feedback helps us write better content