Square Root of 567 — Value, Simplification, and Steps

The square root of 567 is approximately 23.812, and in exact form it is $9\sqrt{7}$, an irrational number that never terminates or repeats.

Quick Answer:

Result: $\sqrt{567} \approx 23.812$

Notation: $\sqrt{567} = 9\sqrt{7}$ (simplest radical form)

Method shown: Prime factorization and long division

Approximate value (irrational): $23.811761800$

Exact form: $9\sqrt{7}$

Quick Reference Table

Where The Square Root of 567 Appears

$\sqrt{567}$ appears in distance problems on a coordinate grid whenever the squared gaps add to $567$, and in any right triangle whose leg-squares sum to $567$. Because $567 = 81 \times 7$, it is also the diagonal of a rectangle whose sides are $9$ and $9\sqrt{6}$, the kind of simplification that keeps surd arithmetic exact instead of drifting into rounding error.

Where The Square Soot of 567 Appears

$\sqrt{567}$ appears in distance problems on a coordinate grid whenever the squared gaps add to $567$, and in any right triangle whose leg-squares sum to $567$. Because $567 = 81 \times 7$, it is also the diagonal of a rectangle whose sides are $9$ and $9\sqrt{6}$, the kind of simplification that keeps surd arithmetic exact instead of drifting into rounding error.

What The Square Root of 567 Means

The square root of 567 is the positive number whose square is $567$. In symbols, $\sqrt{567} \times \sqrt{567} = 567$.

Because $23^2 = 529$ and $24^2 = 576$, the answer lies between $23$ and $24$, close to $24$.

Is the Square Root of 567 Rational or Irrational?

The square root of 567 is irrational. A rational number can be written as a fraction $\frac{p}{q}$ of two integers; $\sqrt{567}$ cannot.

The prime factorization of $567$ is $3^4 \times 7$, so the prime $7$ appears to an odd power. A perfect square needs every prime in an even power, so $567$ is not a perfect square and its root runs on forever without repeating.

How To Compute The Square Root of 567

Method 1: Prime factorization (simplest radical form)

Break $567$ into primes:

$567 = 3 \times 189$

$567 = 3 \times 3 \times 63$

$567 = 3 \times 3 \times 3 \times 21$

$567 = 3^4 \times 7$

Every pair of $3$s comes out as one $3$; two pairs give $3 \times 3 = 9$:

$\sqrt{567} = \sqrt{3^4 \times 7}$

$\sqrt{567} = 9\sqrt{7}$

Since $\sqrt{7} \approx 2.6458$:

$9 \times 2.6458 = 23.812$

Final answer: $\sqrt{567} = 9\sqrt{7} \approx 23.812$

Method 2: Long division

Group the digits of $567$ in pairs from the right: $5$ and $67$.

Find the largest number whose square is at most $5$: that is $2$, since $2^2 = 4$.

Subtract to get remainder $1$, then bring down $67$ to make $167$.

Double the quotient so far ($2$) to get $4$; find a digit $x$ so that $4x \times x \le 167$.

Test $x = 3$: $43 \times 3 = 129 \le 167$; test $x = 4$: $44 \times 4 = 176 > 167$. So $x = 3$.

The quotient is now $23$, remainder $167 - 129 = 38$; place a decimal point and bring down $00$ to make $3800$.

Continue the process to reach $23.81\ldots$

Final answer: $\sqrt{567} \approx 23.812$

Common Mistakes With the Square Root of 567

Mistake 1: Pulling out only one pair of threes

Where it slips in: Factoring $567 = 9 \times 63$ and taking just the $9$.

Don't do this: Write $\sqrt{567} = 3\sqrt{63}$ and stop.

The correct way: $63 = 9 \times 7$ still holds a perfect square. The full factorization is $3^4 \times 7$, giving $9\sqrt{7}$. The first instinct is to stop at the first square factor you spot, but $\sqrt{63}$ still simplifies — keep going until the number under the radical has no square factor.

Mistake 2: Reading 567 as close to 576 and writing 24

Where it slips in: Noticing $567$ is near $576 = 24^2$.

Don't do this: Write $\sqrt{567} = 24$.

The correct way: $24^2 = 576 \ne 567$, so $\sqrt{567} < 24$; the value is $23.812$.

Mistake 3: Miscounting powers of 3

Where it slips in: Deciding how many $3$s come out of the radical.

Don't do this: Treat $3^4$ as giving a single $3$ outside.

The correct way: $3^4 = (3^2)^2$, so $\sqrt{3^4} = 3^2 = 9$. Each pair of equal factors leaves one copy outside.

A Quick Way to Check Yourself

Estimate first: $567$ sits between $529$ and $576$, so the root is between $23$ and $24$, and closer to $24$. Then confirm the surd form by squaring it back: $(9\sqrt{7})^2 = 81 \times 7 = 567$. To build surd 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 153 — simplifies to $3\sqrt{17}$.

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

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

• Squares and square roots — the wider topic.