Book A Free Math Class

Square Root of 153 — Value, Simplification, and Steps

#Algebra

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

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.