What Is A Square Root?
A square root of a number $n$ is a value $r$ that satisfies $r^2 = n$. The square root of 59 is the positive number that, multiplied by itself, gives 59.
No integer does this. Checking the two closest whole numbers:
$$7^2 = 49$$
$$8^2 = 64$$
Since 59 sits between 49 and 64, $\sqrt{59}$ sits between 7 and 8. The relationship between a number and its root is the whole story behind squares and square roots, and here it tells us the answer starts with 7 and a decimal.
Where Does √59 Appear?
$\sqrt{59}$ is the exact side length of a square whose area is 59 square units. Draw a square that holds 59 unit tiles, and each edge measures $\sqrt{59} \approx 7.68$ units, a length that lives between 7 and 8 but never lands on a whole number. It also turns up whenever the Pythagorean theorem produces a leg sum of 59, such as a right triangle with legs of length $\sqrt{35}$ and $\sqrt{24}$, whose hypotenuse is $\sqrt{35 + 24} = \sqrt{59}$.
Quick Reference Table
Number $n$ | $\sqrt{n}$ (approx.) | Simplified | Rational or Irrational |
|---|---|---|---|
49 | 7 | $7$ | Rational |
50 | 7.0711 | $5\sqrt{2}$ | Irrational |
56 | 7.4833 | $2\sqrt{14}$ | Irrational |
59 | 7.6811 | $\sqrt{59}$ | Irrational |
60 | 7.7460 | $2\sqrt{15}$ | Irrational |
61 | 7.8102 | $\sqrt{61}$ | Irrational |
64 | 8 | $8$ | Rational |
For the full pattern from 1 upward, see the reference on square roots from 1 to 30.
Is The Square Root Of 59 Rational Or Irrational?
$\sqrt{59}$ is irrational. It cannot be written as a fraction $\frac{p}{q}$ of two integers, and its decimal expansion neither terminates nor repeats.
The quick reason. A whole number has a rational square root only when it is a perfect square, like 49 or 64. 59 is prime, so it is not a perfect square, and its root cannot be a whole number or a clean fraction. That places $\sqrt{59}$ among the irrational numbers, alongside $\sqrt{2}$ and $\sqrt{3}$.
Why prime seals it. For $\sqrt{59}$ to be rational, 59 would need a repeated prime factor to form a perfect square inside it. Its only factors are 1 and 59, so nothing pairs up. The radical stays as $\sqrt{59}$, and every decimal you write for it is an approximation.
How Do You Find √59? (Long Division And Estimation)
Because 59 is prime, prime factorization gives $59 = 59$ with no pair to extract, so it cannot simplify the radical. Two methods give the decimal instead.
Method 1: Estimation between perfect squares
Find the nearest perfect squares below and above 59.
$$7^2 = 49 \quad \text{and} \quad 8^2 = 64$$
59 is 10 above 49 and 5 below 64, so it sits past the midpoint, closer to 8.
A first estimate: $7 + \frac{59 - 49}{64 - 49} = 7 + \frac{10}{15} \approx 7.67$.
That is already within 0.02 of the true value. For faster mental estimates like this, the square root tricks reference collects the shortcuts.
Method 2: Long division (exact digits)
Step 1: Write 59 and pair zeros after the decimal point.
$$59.\overline{00},\overline{00},\overline{00}$$
Step 2: The largest whole number with a square at most 59 is 7, since $7^2 = 49$.
$$59 - 49 = 10$$
Step 3: Bring down a pair of zeros to get 1000. Double the quotient 7 to get 14.
Step 4: Find $d$ so that $(140 + d) \times d \leq 1000$. Testing $d = 6$: $146 \times 6 = 876$. Quotient becomes 7.6.
$$1000 - 876 = 124$$
Step 5: Bring down zeros to get 12400. Double 76 to get 152.
Step 6: Find $d$ so that $(1520 + d) \times d \leq 12400$. Testing $d = 8$: $1528 \times 8 = 12224$. Quotient becomes 7.68.
$$12400 - 12224 = 176$$
Step 7: Continue two more rounds to reach 7.6811.
$$\sqrt{59} \approx 7.6811$$
The process never ends, which is exactly what "irrational" looks like on paper.
Examples Of √59
Example 1
Estimate $\sqrt{59}$ to the nearest tenth without a calculator.
The nearest perfect squares are $7^2 = 49$ and $8^2 = 64$.
59 is well past the midpoint 56.5, so the root is above 7.5.
Testing $7.7^2 = 59.29$, slightly too high.
Testing $7.68^2 = 58.98$, just under.
Final answer: $\sqrt{59} \approx 7.7$ to the nearest tenth.
Example 2
A common slip worth walking through: simplify $\sqrt{59}$.
The tempting move is to split 59 into $\sqrt{50} \times \sqrt{9}$ because $\sqrt{9} = 3$ looks clean.
Wrong path: $50 \times 9 = 450$, not 59, so that split is false from the start.
Where it breaks: simplifying a radical needs a factor pair of the actual number, and 59 has none.
The rescue: list the factors of 59. They are only 1 and 59. No perfect square greater than 1 divides it.
Final answer: $\sqrt{59}$ is already in simplest form.
Example 3
Solve $x^2 = 59$.
Take the square root of both sides.
$$x = \pm\sqrt{59}$$
$$x \approx \pm 7.6811$$
Final answer: $x = \sqrt{59}$ or $x = -\sqrt{59}$, about $\pm 7.6811$.
Example 4
Find the side of a square whose area is 59 square centimetres.
Area of a square is side squared, so side $= \sqrt{\text{area}}$.
$$\text{side} = \sqrt{59} \approx 7.6811 \text{ cm}$$
Final answer: each side is about 7.68 cm.
Example 5
Evaluate $(\sqrt{59})^2 + 1$.
Squaring undoes the root.
$$(\sqrt{59})^2 = 59$$
$$59 + 1 = 60$$
Final answer: 60.
Common Mistakes
Mistake 1: Forcing a simplification
Where it slips in: students assume every radical reduces to something smaller.
Don't do this: writing $\sqrt{59} = 7\sqrt{...}$ or splitting it into non-factors.
The correct way: check the factors first. 59 is prime, so $\sqrt{59}$ stays as $\sqrt{59}$. A radical only simplifies when the number has a perfect-square factor above 1.
Mistake 2: Rounding to the wrong neighbour
Where it slips in: estimating $\sqrt{59}$ to the nearest whole number. Students meeting this for the first time often anchor to 7 because $7^2 = 49$ is the square they see first.
Don't do this: answering 7 without checking the midpoint.
The correct way: the midpoint between $7^2$ and $8^2$ maps to about 7.5. Since $\sqrt{59} \approx 7.68$ is above that, the nearest whole number is 8.
Mistake 3: Rounding too early in a longer problem
Where it slips in: multi-step problems where $\sqrt{59}$ appears mid-calculation.
Don't do this: replacing $\sqrt{59}$ with 7.68 at the start and carrying that through.
The correct way: keep $\sqrt{59}$ in radical form until the final step, then round once.
Conclusion
The square root of 59 is about 7.6811, irrational, and already in its simplest form because 59 is prime. You can estimate it by squeezing 59 between 49 and 64, or compute exact digits by long division, and the same reasoning handles any non-perfect-square root you meet. To build this skill with a teacher, explore Bhanzu's algebra tutor sessions, get targeted help with algebra, or join structured math classes online. You can also book a free demo class to see the approach first.
Read More
Square Root of 61 — the next prime root above 59, worked the same way.
Square Root of 50 — a nearby root that does simplify, to $5\sqrt{2}$.
Square Root of 64 — the perfect square just above 59.
Square Root 1 to 25 — a quick-lookup table of the smaller roots.
Exponents — how radicals connect to fractional powers.
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