Book A Free Math Class

# Square Root of 153 — Value, Simplification, and Steps

[#Algebra](/content/tag/algebra/index.html)

TL;DR  
The square root of 153 is 3173\sqrt{17}317​ in exact form and approximately 12.369 as a decimal. This article gives the value, shows why 153\sqrt{153}153​ is irrational, simplifies it by prime factorization, and works the long-division method step by step.

The square root of 153 is approximately **12.369**, and in exact form it is 3173\sqrt{17}317​, an irrational number that never terminates or repeats.

> **Quick Answer:**  
> **Result:** 153≈12.369\sqrt{153} \approx 12.369153​≈12.369  
> **Notation:** 153=317\sqrt{153} = 3\sqrt{17}153​=317​ (simplest radical form)  
> **Method shown:** Prime factorization and long division  
> **Approximate value (irrational):** 12.369316877  
> **Exact form:** 3173\sqrt{17}317​

## Quick Reference Table  
| Number | Square root (approx.) | Exact / simplified | Rational? |  
| --- | --- | --- | --- |  
| 43 | 6.557 | 43\sqrt{43}43​ | Irrational |  
| 150 | 12.247 | 565\sqrt{6}56​ | Irrational |  
| 153 | 12.369 | 3173\sqrt{17}317​ | Irrational |  
| 156 | 12.490 | 2392\sqrt{39}239​ | Irrational |  
| 169 | 13.000 | 131313 | Rational |  
| 255 | 15.969 | 255\sqrt{255}255​ | Irrational |  
| 567 | 23.812 | 979\sqrt{7}97​ | Irrational |

## Where The Square Root of 153 Appears  
153\sqrt{153}153​ is the length of the diagonal of a rectangle with sides 33 and 1212, since 32+122=9+144=1533^2 + 12^2 = 9 + 144 = 15332+122=9+144=153. It also shows up whenever a right triangle has legs whose squares sum to 153153153, and in distance calculations on a coordinate grid where the horizontal and vertical gaps square to 153153153.

## What The Square Root of 153 Means  
The **square root of 153** is the positive number that, multiplied by itself, gives 153153153. In symbols, 153×153=153\sqrt{153} \times \sqrt{153} = 153153​×153​=153.

Because 122=14412^2 = 144122=144 and 132=16913^2 = 169132=169, the answer sits between 121212 and 131313, closer to 121212.

### Is the square root of 153 rational or irrational?  
The square root of 153 is **irrational**. A number is rational only if it can be written as a fraction pq\frac{p}{q}qp​ of two integers.

The prime factorization of 153153153 is 3×3×173 \times 3 \times 173×3×17, so 171717 appears to an odd power. A perfect square needs every prime to appear an even number of times, so 153153153 is not a perfect square and its root cannot be an exact fraction.

## How to compute the square root of 153

### **Method 1: Prime factorization (simplest radical form)**  
Break 153153153 into primes:  
153=3×51153 = 3 \times 51153=3×51  
153=3×3×17153 = 3 \times 3 \times 17153=3×3×17  
153=32×17153 = 3^2 \times 17153=32×17

Pull the pair of 333s out of the radical:  
153=32×17\sqrt{153} = \sqrt{3^2 \times 17}153​=32×17​

Since 17≈4.123\sqrt{17} \approx 4.12317​≈4.123:  
3×4.123=12.3693 \times 4.123 = 12.3693×4.123=12.369  
**Final answer:** 153=317≈12.369\sqrt{153} = 3\sqrt{17} \approx 12.369153​=317​≈12.369

### **Method 2: Long division**  
Group the digits of 153153153 in pairs from the right: 111 and 535353.  
Find the largest number whose square is at most 111: that is 111, since 12=11^2 = 112=1.  
Subtract to get remainder 000, then bring down 535353 to make 535353.  
Double the quotient so far (111) to get 222; find a digit xxx so that 2x×x≤532x \times x \le 532x×x≤53.

Test x=2x = 2x=2: 22×2=44≤5322 \times 2 = 44 \le 5322×2=44≤53; test x=3x = 3x=3: 23×3=69>5323 \times 3 = 69 > 5323×3=69>53. So x=2x = 2x=2.  
The quotient is now 121212, remainder 53−44=953 - 44 = 953−44=9; place a decimal point and bring down 000000 to make 900900900.

Continue the process to reach 12.36…12.36\ldots12.36…  
**Final answer:** 153≈12.369\sqrt{153} \approx 12.369153​≈12.369

## Common Mistakes With The Square Root of 153  
### **Mistake 1: Calling 153 a perfect square**  
**Where it slips in:** Seeing that 153153153 is close to 144=122144 = 12^2144=122 and rounding to a whole number.  
**Don't do this:** Write 153=12\sqrt{153} = 12153​=12.  
**The correct way:** 122=144≠15312^2 = 144 \ne 153122=144=153, so the root is irrational; leave it as 3173\sqrt{17}317​ or approximate to 12.36912.36912.369. The first instinct is to snap to the nearest whole number, but 153153153 lands between two perfect squares, not on one.

### **Mistake 2: Stopping the simplification too early**  
**Where it slips in:** Factoring 153153153 as 9×179 \times 179×17 and leaving it under the radical.  
**Don't do this:** Write 153=9×17\sqrt{153} = \sqrt{9 \times 17}153​=9×17​ and stop.  
**The correct way:** 9=3\sqrt{9} = 39​=3 comes out of the radical, giving 3173\sqrt{17}317​. A perfect-square factor left inside is a half-finished answer.

### **Mistake 3: Pairing digits from the left in long division**  
**Where it slips in:** Grouping 153153153 as 151515 and 333.  
**Don't do this:** Pair from the left.  
**The correct way:** Always pair from the right: 111 and 535353. Pairing the wrong way throws off every later step.

## A Quick Way To Check Yourself  
Estimate before you compute: since 153153153 sits between 144144144 and 169169169, the root must sit between 121212 and 131313, so any answer far outside that range is a mistake.

## Frequently Asked Questions  
**What is the square root of 153 in simplest radical form?**  
It is 3173\sqrt{17}317​, because 153=32×17153 = 3^2 \times 17153=32×17 and the pair of threes comes out of the radical.

**Is the square root of 153 rational or irrational?**  
Irrational. 153153153 is not a perfect square, so its root cannot be written as an exact fraction and its decimal never ends.

**What is the square root of 153 to three decimal places?**  
153≈12.369\sqrt{153} \approx 12.369153​≈12.369.

**What is the square root of 153 as a decimal?**  
Approximately 12.369316877, continuing without any repeating pattern.

**Is 153 a perfect square?**  
No. The nearest perfect squares are 144=122144 = 12^2144=122 and 169=132169 = 13^2169=132, and 153153153 falls between them.