Square Root of 148 - How to Find the Square Root of 148?

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

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.