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 148 is the number that, multiplied by itself, gives 148.
No integer does this. Since $12^2 = 144$ is just under and $13^2 = 169$ is over, $\sqrt{148}$ lies between 12 and 13, just above 12.
The symbol $\sqrt{148}$ means the principal (positive) root, even though $(-12.1655)^2$ also equals 148.
Where Does √148 Appear?
$\sqrt{148}$ is the side length of a square with an area of 148 square units, a value just above 12. It also appears through the Pythagorean theorem whenever two squared legs add to 148 - for example, a right triangle with legs 12 and 2, since $144 + 4 = 148$.
Quick Reference Table
Number $n$ | $\sqrt{n}$ (approx.) | Simplest radical | Rational or Irrational |
|---|---|---|---|
144 | 12 | 12 | Rational |
148 | 12.1655 | $2\sqrt{37}$ | Irrational |
150 | 12.2474 | $5\sqrt{6}$ | Irrational |
153 | 12.3693 | $3\sqrt{17}$ | Irrational |
160 | 12.6491 | $4\sqrt{10}$ | Irrational |
169 | 13 | 13 | Rational |
Is the Square Root of 148 Rational or Irrational?
$\sqrt{148}$ is irrational - it cannot be written as a fraction of two integers, and its decimal never terminates or repeats.
Why? A whole number has a rational square root only when it is a perfect square. 148 is not a perfect square, so $\sqrt{148}$ is irrational even after simplifying to $2\sqrt{37}$.
Can the square root of 148 be simplified? Yes. 148 carries the perfect-square factor 4, so it reduces to $2\sqrt{37}$ — the same reduction step practised across simplifying radical expressions.
How Do You Find √148? (Prime Factorization and Long Division)
Prime factorization performs the simplification.
$$148 = 2 \times 2 \times 37$$
$$148 = 2^2 \times 37$$
The pair of 2s is a perfect square, so a 2 leaves the radical.
$$\sqrt{148} = \sqrt{2^2 \times 37} = 2\sqrt{37}$$
Because 37 is prime, nothing else comes out. The square root 1 to 30 hub shows which nearby roots reduce and which stay put.
Long division produces the decimal. Pair the digits: $1,48.\overline{00},\overline{00}$.
Step 1: Find the largest integer whose square is $\leq 148$.
$$12^2 = 144 \leq 148$$
Step 2: Subtract and bring down the first pair of zeros.
$$148 - 144 = 4$$
$$\text{new dividend} = 400$$
Step 3: Double the quotient 12 to get 24, then find a digit $d$ with $(240 + d) \times d \leq 400$.
$$241 \times 1 = 241$$
Step 4: Subtract and bring down the next pair.
$$400 - 241 = 159$$
$$\text{new dividend} = 15900$$
Step 5: Double 121 to get 242, then find $d$ with $(2420 + d) \times d \leq 15900$.
$$2426 \times 6 = 14556$$
Step 6: One more round gives:
$$\sqrt{148} \approx 12.1655$$
For a quick estimate without long division, the nearest-perfect-square method in square root tricks places $\sqrt{148}$ just above $\sqrt{144} = 12$.
Examples of √148
Example 1
Confirm that $\sqrt{148} = 2\sqrt{37}$ numerically.
$$2\sqrt{37} = 2 \times 6.0828 = 12.1655$$
This matches the long-division value.
Final answer: $\sqrt{148} = 2\sqrt{37} \approx 12.1655$.
Example 2
Rewrite $2\sqrt{37}$ as a single radical.
The tempting move is to slide the 2 inside next to the 37:
$$2\sqrt{37} = \sqrt{74} \quad ?$$
Check it. $\sqrt{74} \approx 8.60$, but $2\sqrt{37} \approx 12.17$. They do not match.
The slip is failing to square the coefficient before it enters the radical. A number moving inside a square root must be squared, so $2\sqrt{37} = \sqrt{2^2 \times 37} = \sqrt{148}$.
Final answer: $2\sqrt{37} = \sqrt{148}$, not $\sqrt{74}$.
Example 3
Between which two consecutive integers does $\sqrt{148}$ lie?
$$12^2 = 144$$
$$13^2 = 169$$
Since $144 < 148 < 169$, the root sits between 12 and 13, close to 12 because 148 is near 144.
Final answer: between 12 and 13, near 12.17.
Example 4
A square garden covers 148 square metres. What is its side, in simplest radical form?
Side length $= \sqrt{\text{area}}$.
$$\text{side} = \sqrt{148} = 2\sqrt{37} \approx 12.17 \text{ m}$$
Final answer: $2\sqrt{37}$ m, about $12.17$ m.
Example 5
Evaluate $\sqrt{148} \times \sqrt{37}$.
Write $\sqrt{148}$ in simplest form first.
$$\sqrt{148} \times \sqrt{37} = 2\sqrt{37} \times \sqrt{37}$$
$$= 2 \times 37 = 74$$
Final answer: 74.
Common Mistakes
Mistake 1: Moving the coefficient inside without squaring it
Where it slips in: When rewriting $2\sqrt{37}$ as a single radical.
Don't do this: Writing $2\sqrt{37} = \sqrt{74}$.
The correct way: Square the coefficient first: $2\sqrt{37} = \sqrt{4 \times 37} = \sqrt{148}$. Forgetting to square the number as it crosses into the radical is a habit that a quick reverse-check catches every time.
Mistake 2: Stopping the simplification too soon
Where it slips in: When a student factors $148 = 2 \times 74$ and writes $\sqrt{2}\sqrt{74}$.
Don't do this: Leaving $\sqrt{2}\sqrt{74}$, unaware that 74 still splits into $2 \times 37$.
The correct way: Factor all the way to primes to reveal $2^2 \times 37$, giving $2\sqrt{37}$.
Mistake 3: Calling 148 a perfect square
Where it slips in: When a quick glance near 144 suggests a whole-number root.
Don't do this: Writing $\sqrt{148} = 12$.
The correct way: $12^2 = 144$, not 148, so $\sqrt{148}$ is irrational and equals $2\sqrt{37}$.
Conclusion
The square root of 148 simplifies to $2\sqrt{37}$, roughly $12.1655$, and stays irrational because 148 is not a perfect square. To practise radical simplification with a teacher, explore Bhanzu's algebra tutor or a high school math tutor, or join structured math classes online. Want the factor-tree method worked live? Book a free demo class.
Read More
Square Root of 20 — reduces to $2\sqrt{5}$ by the same coefficient step.
Square Root of 50 — simplifies to $5\sqrt{2}$, another perfect-square extraction.
Square Root of 153 — a nearby root that reduces to $3\sqrt{17}$.
Squares and Square Roots — the full relationship between a number and its root.
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