What Is A Square Root?
The square root of a number $n$ is the value $r$ for which $r^2 = n$, the number that, multiplied by itself, returns $n$. So $\sqrt{720}$ is the number whose square is 720.
No whole number fits: $26^2 = 676$ is short and $27^2 = 729$ overshoots by only 9. That places $\sqrt{720}$ between 26 and 27, close to 27.
Where Does √720 Appear?
$\sqrt{720}$ is the natural answer to a distance question: a right triangle with legs of $12$ and $24$ has a hypotenuse of $\sqrt{12^2 + 24^2} = \sqrt{144 + 576} = \sqrt{720}$, or $12\sqrt{5}$. It also shows up in area work, where the side of a square that covers 720 square units measures $12\sqrt{5}$ units, just under $26.84$, since $720$ is $144 \times 5$ and 144 is a clean perfect square.
Quick Reference Table
Number $n$ | $\sqrt{n}$ (simplified) | $\sqrt{n}$ (approx.) |
|---|---|---|
676 | 26 | 26 |
700 | $10\sqrt{7}$ | 26.4575 |
720 | $\mathbf{12\sqrt{5}}$ | 26.8328 |
729 | 27 | 27 |
750 | $5\sqrt{30}$ | 27.3861 |
800 | $20\sqrt{2}$ | 28.2843 |
Is 720 A Perfect Square?
No. A perfect square is an integer multiplied by itself, $1, 4, 9, 16, 25, \ldots, 676, 729$, and 720 is not on that list.
Because 720 is not a perfect square, its square root is not a whole number or a fraction. That is what makes $\sqrt{720}$ irrational, even though a large perfect square, 144, hides inside it.
Is The Square Root Of 720 Rational Or Irrational?
$\sqrt{720}$ is irrational, it cannot be written as a fraction $\frac{p}{q}$ of integers, and its decimal neither ends nor repeats. A whole number has a rational square root only when it is a perfect square, and 720 is not.
Simplifying to $12\sqrt{5}$ does not change this. The leftover $\sqrt{5}$ is irrational, so the whole product stays irrational, which follows from the formal definition of a square root.
What Is The Square Root Of 720 In Simplest Radical Form?
The simplest radical form of $\sqrt{720}$ is $12\sqrt{5}$. For a square root you pull out factors that appear twice, the same grouping logic used in simplifying radical expressions.
Prime factorization method.
$720 = 2 \times 2 \times 2 \times 2 \times 3 \times 3 \times 5$
$720 = 2^4 \times 3^2 \times 5$
$\sqrt{720} = \sqrt{2^4 \times 3^2 \times 5}$
$\sqrt{720} = 2^2 \times 3 \times \sqrt{5}$
$\sqrt{720} = 12\sqrt{5}$
The four 2s make two pairs, giving $2^2 = 4$ outside, and the two 3s make one pair, giving a single 3 outside. Together $4 \times 3 = 12$ steps out, while the lone 5 has no partner and stays inside as $\sqrt{5}$. The leftover radical is exactly the square root of 5, which is why $\sqrt{720}$ belongs to the same family as the square root of 20, or $2\sqrt{5}$.
How Do You Find √720 By Long Division?
Prime factorization gives the exact form; long division gives the decimal digit by digit. Here it is, one step per line, using the square root tricks that work without a calculator.
Pair the digits from the decimal point: $7\ \overline{20}.\overline{00}\ \overline{00}$
The largest square not exceeding 7 is $4 = 2^2$, so the first digit is 2, remainder 3.
Bring down 20 to get 320; double the quotient to get 4, and find $d$ with $(40 + d)\times d \le 320$; $46 \times 6 = 276$, so the next digit is 6, quotient 26.
Subtract to get remainder 44; bring down a pair of zeros to get 4400; double 26 to get 52, and find $d$ with $(520 + d)\times d \le 4400$; $528 \times 8 = 4224$, so the next digit is 8, quotient $26.8$.
Continue the same way and the digits settle at $\sqrt{720} \approx 26.8328$. The value never repeats, which is the signature of an irrational number. A quick sanity path: since 720 sits between $26^2 = 676$ and $27^2 = 729$, and only 9 short of 729, the root should land just under 27, which it does.
Examples Of √720
Example 1
Simplify $\sqrt{720}$ to its radical form.
$720 = 2^4 \times 3^2 \times 5$
$\sqrt{720} = 2^2 \times 3 \times \sqrt{5} = 12\sqrt{5}$
Final answer: $12\sqrt{5}$
Example 2
A student simplifies $\sqrt{720}$ and writes $\sqrt{720} = 6\sqrt{20}$, then stops. Where does it go wrong?
The first move is fine: $720 = 36 \times 20$, so $\sqrt{720} = 6\sqrt{20}$. The problem is stopping there, because 20 still hides a perfect square, $20 = 4 \times 5$.
Push the simplification all the way: $6\sqrt{20} = 6 \times 2\sqrt{5} = 12\sqrt{5}$. A radical is simplest only when nothing square remains inside.
Final answer: $12\sqrt{5}$
Example 3
Evaluate $(\sqrt{720})^2$.
$(\sqrt{720})^2 = 720$
Final answer: $720$
Example 4
A square plot has an area of 720 square metres. Find its side length.
side $= \sqrt{720}$
side $= 12\sqrt{5} \approx 26.8328$
Final answer: $12\sqrt{5}$ m, about $26.83$ m.
Example 5
Simplify $\sqrt{720} - \sqrt{80}$.
$\sqrt{720} - \sqrt{80} = 12\sqrt{5} - 4\sqrt{5}$
$\sqrt{720} - \sqrt{80} = 8\sqrt{5} \approx 17.8885$
Final answer: $8\sqrt{5}$. Both roots reduce to a multiple of $\sqrt{5}$, so they subtract like terms once each is in simplest form.
Common Mistakes
Mistake 1: Stopping before the radical is simplest
Where it slips in: Factoring out a partial perfect square and quitting early.
Don't do this: Writing $\sqrt{720} = 6\sqrt{20}$ or $4\sqrt{45}$ as the final answer.
The correct way: Both $20$ and $45$ still contain the perfect square factors 4 and 9. Reduce fully to $12\sqrt{5}$, or factor out the largest perfect square, 144, in one step.
Mistake 2: Mishandling the odd prime
Where it slips in: Trying to pull the lone 5 out of the radical.
Don't do this: Writing $\sqrt{720} = 12 \times 5 = 60$ or $\sqrt{720} = 60\sqrt{5}$.
The correct way: The 5 appears only once, so it has no pair and cannot leave. It stays inside as $\sqrt{5}$, giving $12\sqrt{5}$, and the decimal $26.83$ confirms it, well below 60.
Mistake 3: Rounding too early
Where it slips in: Multi-step problems where $\sqrt{720}$ appears mid-calculation.
Don't do this: Replacing $\sqrt{720}$ with $26.83$ at the start and carrying that value through every step.
The correct way: Keep $12\sqrt{5}$ in exact form until the final line. Early rounding introduces error that compounds across multiplications and divisions.
Conclusion
The square root of 720 is $12\sqrt{5}$, roughly $26.8328$, and it is irrational because 720 has no whole-number square root. Write $720 = 2^4 \times 3^2 \times 5$, send out one factor for each pair, and keep the lone 5 inside. To build these radical and exponent skills with a teacher, explore Bhanzu's algebra tutor, work with a high school math tutor, or join live math classes online.
Want to practice with a guide? Book a free demo class and work through radicals step by step.
Read More
Square Root 1 to 30, the full reference table of square roots from 1 to 30.
Square Root of 360, the root of half of 720, which reduces to $6\sqrt{10}$.
Square Root of 50, a clean pairing example that reduces to $5\sqrt{2}$.
Squares and Square Roots, how roots and powers connect across the number line.
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