Square Root of 153 — Value, Simplification, and Steps

The square root of 153 is approximately 12.369, and in exact form it is $3\sqrt{17}$, an irrational number that never terminates or repeats.

Quick Answer:

Result: $\sqrt{153} \approx 12.369$

Notation: $\sqrt{153} = 3\sqrt{17}$ (simplest radical form)

Method shown: Prime factorization and long division

Approximate value (irrational): $12.369316877$

Exact form: $3\sqrt{17}$

Quick Reference Table

Where The Square Root of 153 Appears

$\sqrt{153}$ is the length of the diagonal of a rectangle with sides $3$ and $12$, since $3^2 + 12^2 = 9 + 144 = 153$. It also shows up whenever a right triangle has legs whose squares sum to $153$, and in distance calculations on a coordinate grid where the horizontal and vertical gaps square to $153$.

What The Square Root of 153 Means

The square root of 153 is the positive number that, multiplied by itself, gives $153$. In symbols, $\sqrt{153} \times \sqrt{153} = 153$.

Because $12^2 = 144$ and $13^2 = 169$, the answer sits between $12$ and $13$, closer to $12$.

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 $\frac{p}{q}$ of two integers.

The prime factorization of $153$ is $3 \times 3 \times 17$, so $17$ appears to an odd power. A perfect square needs every prime to appear an even number of times, so $153$ 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 $153$ into primes:

$153 = 3 \times 51$

$153 = 3 \times 3 \times 17$

$153 = 3^2 \times 17$

Pull the pair of $3$s out of the radical:

$\sqrt{153} = \sqrt{3^2 \times 17}$

$\sqrt{153} = 3\sqrt{17}$

Since $\sqrt{17} \approx 4.123$:

$3 \times 4.123 = 12.369$

Final answer: $\sqrt{153} = 3\sqrt{17} \approx 12.369$

Method 2: Long division

Group the digits of $153$ in pairs from the right: $1$ and $53$.

Find the largest number whose square is at most $1$: that is $1$, since $1^2 = 1$.

Subtract to get remainder $0$, then bring down $53$ to make $53$.

Double the quotient so far ($1$) to get $2$; find a digit $x$ so that $2x \times x \le 53$.

Test $x = 2$: $22 \times 2 = 44 \le 53$; test $x = 3$: $23 \times 3 = 69 > 53$. So $x = 2$.

The quotient is now $12$, remainder $53 - 44 = 9$; place a decimal point and bring down $00$ to make $900$.

Continue the process to reach $12.36\ldots$

Final answer: $\sqrt{153} \approx 12.369$

Common Mistakes With The Square Root of 153

Mistake 1: Calling 153 a perfect square

Where it slips in: Seeing that $153$ is close to $144 = 12^2$ and rounding to a whole number.

Don't do this: Write $\sqrt{153} = 12$.

The correct way: $12^2 = 144 \ne 153$, so the root is irrational; leave it as $3\sqrt{17}$ or approximate to $12.369$. The first instinct is to snap to the nearest whole number, but $153$ lands between two perfect squares, not on one.

Mistake 2: Stopping the simplification too early

Where it slips in: Factoring $153$ as $9 \times 17$ and leaving it under the radical.

Don't do this: Write $\sqrt{153} = \sqrt{9 \times 17}$ and stop.

The correct way: $\sqrt{9} = 3$ comes out of the radical, giving $3\sqrt{17}$. 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 $153$ as $15$ and $3$.

Don't do this: Pair from the left.

The correct way: Always pair from the right: $1$ and $53$. Pairing the wrong way throws off every later step.

A Quick Way To Check Yourself

Estimate before you compute: since $153$ sits between $144$ and $169$, the root must sit between $12$ and $13$, so any answer far outside that range is a mistake. To build square-root fluency with a teacher, explore Bhanzu's algebra tutor or structured help with algebra.

Read More

• Square root — what a square root is and how the symbol works.

• Square root 1 to 25 — the reference chart for the small roots.

• Square root of 43 — a prime, irrational root.

• Square root of 255 — irrational with no square factor.

• Square root of 567 — simplifies to $9\sqrt{7}$.

• Squares and square roots — the wider topic.