Square Root of 296 - How to Simplify and Find √296?

What Is A Square Root?

The square root of a number $n$ is the value $r$ with $r^2 = n$. For $296$, no integer fits, because $17^2 = 289$ is too small and $18^2 = 324$ is too big, so $\sqrt{296}$ falls between $17$ and $18$.

To simplify any square root, you look for the largest perfect square dividing the number, a habit that runs through all of squares and square roots. For $296$, that perfect square is $4$, which is what lets $\sqrt{296}$ shrink to $2\sqrt{74}$.

Where Does √296 Appear?

$\sqrt{296}$ is the diagonal of a $10 \times 14$ rectangle, because the Pythagorean theorem gives $\sqrt{10^2 + 14^2} = \sqrt{100 + 196} = \sqrt{296}$. It is also the distance between the points $(0, 0)$ and $(10, 14)$ on a coordinate plane, which is where a number like $2\sqrt{74}$ stops being abstract. Any time a right triangle has legs $10$ and $14$, its longest side measures exactly $2\sqrt{74} \approx 17.20$ units.

Quick Reference Table

The rows around $296$ show how close it sits to the perfect squares $289$ and $324$, and how it simplifies.

Is The Square Root Of 296 Rational Or Irrational?

$\sqrt{296}$ is irrational. It cannot be written as a fraction $\frac{p}{q}$ of integers, and its decimal neither stops nor settles into a repeating pattern.

The prime factorization makes it obvious:

$$296 = 2^3 \times 37$$

A number is a perfect square only when every prime appears an even number of times. Here the prime $2$ appears three times and $37$ appears once, both odd, so $296$ is not a perfect square, and $\sqrt{296}$ is irrational.

How Do You Simplify And Find √296?

What is the square root of 296 in simplest radical form? Prime factorize, then take out every complete pair.

Begin with the factorization.

$$296 = 2^3 \times 37$$

Split off a pair of $2$s, since a pair leaves the radical as one factor.

$$296 = (2^2) \times (2 \times 37)$$

The first bracket is the perfect square $4$, and the second bracket is $74$.

$$\sqrt{296} = \sqrt{4} \times \sqrt{74}$$

$$\sqrt{296} = 2\sqrt{74}$$

Since $74 = 2 \times 37$ has no perfect-square factor, $2\sqrt{74}$ is fully simplified. For the decimal, use $\sqrt{74} \approx 8.6023$.

$$\sqrt{296} = 2 \times 8.6023$$

$$\sqrt{296} \approx 17.2047$$

The same largest-square-factor idea powers the general method for simplifying radical expressions.

Examples Of √296

Example 1

Simplify $\sqrt{296}$ using its largest perfect-square factor.

The largest perfect square dividing $296$ is $4$.

$$\sqrt{296} = \sqrt{4 \times 74}$$

$$\sqrt{296} = 2\sqrt{74}$$

Example 2

A student rounds $\sqrt{296}$ down to the nearest perfect square. Why is that wrong?

The tempting move is to notice $289 = 17^2$ sits just below $296$ and write:

$$\sqrt{296} = 17$$

Test it: $17^2 = 289$, not $296$, so $17$ is too small.

The true value lies between $17$ and $18$, and the exact form is $2\sqrt{74} \approx 17.20$. A nearby perfect square helps you estimate, but it is not the answer.

Example 3

Evaluate $\sqrt{296}$ to two decimal places.

$$\sqrt{296} = 2\sqrt{74}$$

$$\sqrt{296} \approx 2 \times 8.6023$$

$$\sqrt{296} \approx 17.20$$

Example 4

Find the diagonal of a rectangle with sides $10$ and $14$.

$$d = \sqrt{10^2 + 14^2}$$

$$d = \sqrt{100 + 196}$$

$$d = \sqrt{296} = 2\sqrt{74} \approx 17.20$$

Common Mistakes

Mistake 1: Confusing 296 with the nearest perfect square

Where it slips in: Spotting that $289 = 17^2$ is close.

Don't do this: Reporting $\sqrt{296} = 17$.

The correct way: Use $17$ only as an estimate. The exact value is $2\sqrt{74} \approx 17.20$, which is larger than $17$.

Mistake 2: Choosing a factor that is not a perfect square

Where it slips in: Breaking $296$ as $8 \times 37$ and pulling the $8$.

Don't do this: Writing $\sqrt{296} = 8\sqrt{37}$.

The correct way: Only a perfect square comes out whole. Use $296 = 4 \times 74$, and $\sqrt{4} = 2$, giving $2\sqrt{74}$.

Mistake 3: Splitting the sum under the radical

Where it slips in: When $296$ appears as $10^2 + 14^2$ in a distance problem.

Don't do this: Writing $\sqrt{10^2 + 14^2} = 10 + 14$.

The correct way: Add first, then take the root: $\sqrt{100 + 196} = \sqrt{296} = 2\sqrt{74}$. The 1991 Patriot missile failure at Dhahran came from small rounding errors compounding, a reminder that shortcuts with numbers carry real stakes.

Conclusion

The square root of 296 is $2\sqrt{74}$, about $17.2047$, and the entire job is spotting the perfect square $4$ inside $296$. The value stays irrational because its prime factorization $2^3 \times 37$ carries odd powers, and $74$ cannot shrink any further.

To practise radical simplification with guidance, explore Bhanzu's algebra tutor or help with algebra, and see how sessions run through math tutoring. You can also book a free demo class to walk a few roots end to end.

Read More

• Square root 1 to 30 — the reference table of the roots you meet most often.

• Square root tricks — quick ways to estimate and simplify roots by hand.

• Square root of 8 — a neighbour that simplifies the same way, to $2\sqrt{2}$.

• Square root of 50 — another two-step simplification, equal to $5\sqrt{2}$.

• Square root 1 to 25 — a compact chart for the smaller roots.

• Irrational numbers — why non-perfect-square roots never terminate.