Quick Reference Table
Number $n$ | $\sqrt{n}$ (approx.) | Rational or Irrational |
|---|---|---|
121 | 11 | Rational |
144 | 12 | Rational |
146 | 12.083 | Irrational |
150 | 12.247 | Irrational |
153 | 12.369 | Irrational |
157 | 12.530 | Irrational |
169 | 13 | Rational |
Where Does √146 Appear?
$\sqrt{146}$ is the length of the diagonal of a $5 \times 11$ rectangle, because $5^2 + 11^2 = 25 + 121 = 146$, so the Pythagorean theorem gives a diagonal of $\sqrt{146}$. It also shows up whenever a distance calculation lands on a sum of squares that equals 146, such as the gap between the grid points $(0,0)$ and $(5,11)$ on a coordinate plane.
What Is A Square Root?
A square root of a number $n$ is a value $r$ such that $r^2 = n$. So the square root of 146 is the number that, multiplied by itself, gives 146.
No whole number does this: $12^2 = 144$ is too small and $13^2 = 169$ is too big. So $\sqrt{146}$ lands between 12 and 13, much closer to 12.
Is The Square Root Of 146 Rational Or Irrational?
$\sqrt{146}$ is irrational - it cannot be written as a fraction $\frac{p}{q}$ of two integers, and its decimal never terminates or repeats.
Here is the quick reason. A whole number has a rational square root only when it is a perfect square, and 146 sits strictly between the perfect squares $144$ and $169$.
The prime factorisation confirms it: $146 = 2 \times 73$. Both primes appear to an odd power, so no factor can be pulled out as a whole number, which is exactly what marks an irrational number. The same test explains every entry in the square root 1 to 30 reference list.
How Do You Find √146? (Long Division Method)
The long-division method builds the decimal one digit at a time. It is the same idea as long division with numbers, applied to a square root.
Step 1: Pair the digits around the decimal point. $$1\ \overline{46}.\ \overline{00}\ \overline{00}$$
Step 2: The largest number whose square is at most 1 is 1. $$1^2 = 1$$
Step 3: Subtract and bring down the next pair. $$1 - 1 = 0$$ $$\text{bring down } 46 \rightarrow 46$$
Step 4: Double the quotient (1) to get 2, then find a digit $d$ with $(20 + d)\times d \le 46$. $$22 \times 2 = 44 \le 46$$
Step 5: The quotient is now 12, remainder 2. $$146 - 144 = 2$$
Step 6: Bring down a pair of zeros and continue. $$\text{next digits give } 12.0,\ 12.08,\ 12.083$$
After three decimal places, $\sqrt{146} \approx 12.083$. The process never ends, which is what makes the value irrational.
Examples Of √146
Example 1
Estimate $\sqrt{146}$ to the nearest tenth without a calculator.
Find the two perfect squares around 146. $$12^2 = 144$$ $$13^2 = 169$$
146 is only 2 above 144, so the root is just above 12. $$\sqrt{146} \approx 12.1$$
Example 2
A student writes $\sqrt{146} = \sqrt{100} + \sqrt{46} = 10 + 6.78 = 16.78$. Where does this break?
Watch the wrong path first. Splitting a square root across addition treats $\sqrt{a + b}$ as $\sqrt{a} + \sqrt{b}$.
Test it against a known value. $$\sqrt{100} = 10$$ $$\sqrt{146} \approx 12.08$$
If the split were valid, $\sqrt{146}$ would be about 16.78, larger than $\sqrt{169} = 13$. That is impossible, since 146 is less than 169.
The correct rule: square roots do not distribute over addition. Only $\sqrt{a \times b} = \sqrt{a}\times\sqrt{b}$ holds, so $\sqrt{146}$ stays as $\sqrt{146}$.
Example 3
Simplify $\sqrt{146}$ into simplest radical form.
Factor 146 into primes. $$146 = 2 \times 73$$
Neither prime repeats, so there is no perfect-square factor to pull out. $$\sqrt{146} = \sqrt{146}$$
The answer is already in simplest radical form - the same simplifying radical expressions rule that turns $\sqrt{242}$ into $11\sqrt{2}$ simply finds nothing to pull out here.
Example 4
Find the value of $(\sqrt{146})^2$.
Squaring undoes the square root. $$(\sqrt{146})^2 = 146$$
The answer is exactly 146, with no rounding.
Example 5
A square patch of land has an area of 146 square metres. What is the side length?
The side of a square equals the square root of its area. $$s = \sqrt{146}$$ $$s \approx 12.08 \text{ m}$$
So each side is about 12.08 metres.
Common Mistakes
Mistake 1: Splitting the root across addition
Where it slips in: Trying to shortcut $\sqrt{146}$ using nearby round numbers.
Don't do this: Writing $\sqrt{146} = \sqrt{100} + \sqrt{46}$.
The correct way: Roots split only over multiplication, so keep $\sqrt{146}$ whole or estimate it directly.
Mistake 2: Claiming √146 can be simplified
Where it slips in: Assuming every root reduces to something like $a\sqrt{b}$.
Don't do this: Writing $\sqrt{146} = \sqrt{2}\times\sqrt{73}$ and calling it simpler.
The correct way: Since $146 = 2 \times 73$ has no repeated prime, $\sqrt{146}$ is already simplest — the many learners who expect a clean radical here are the ones who stall.
Mistake 3: Rounding too early
Where it slips in: Multi-step problems that use $\sqrt{146}$ before the final line.
Don't do this: Replacing $\sqrt{146}$ with 12.08 at the start and carrying that through.
The correct way: Keep the radical symbolic until the last step, then round once.
Conclusion
The square root of 146 is about 12.083 and is irrational.
$146 = 2 \times 73$, so $\sqrt{146}$ has no simpler radical form.
Long division gives the decimal digit by digit: $12,\ 12.0,\ 12.08,\ 12.083$.
Roots split over multiplication, never over addition — the trap in Example 2.
To go further with a teacher, explore Bhanzu's algebra tutor sessions, a high school math tutor, or live math classes online. Want to see the method taught live? Book a free demo class.
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