Square Root of 87 - Value, Radical Form, and How to Find It

What Is a Square Root?

The square root of a number $n$ is the value $r$ for which $r^2 = n$. So $\sqrt{87}$ is the number that, multiplied by itself, gives 87.

No whole number does this: $9^2 = 81$ is too small and $10^2 = 100$ is too big. That places $\sqrt{87}$ between 9 and 10, near the low end.

Where Does the Square Root of 87 Appear?

Because $\sqrt{87}$ does not tidy into a smaller radical, it usually appears as a raw measurement rather than a neat textbook value. It is the distance from the origin to the point $(9, \sqrt{6})$, since $9^2 + (\sqrt{6})^2 = 81 + 6 = 87$, and it works as a handy estimation target, a value that sits just under $9.33$, close enough to 9 that mental checks stay quick.

Quick Reference Table

Is 87 a Perfect Square?

No. A perfect square is an integer times itself, 1, 4, 9, 16, 25, and onward, and 87 is not on that list.

Since 87 is not a perfect square, its root is not a whole number or a fraction. That is what makes $\sqrt{87}$ irrational, a point you can explore further in the reference definition of a square root.

Is the Square Root of 87 Rational or Irrational?

$\sqrt{87}$ is irrational, it cannot be written as a fraction $\frac{p}{q}$ of integers, and its decimal neither ends nor repeats. A whole number has a rational square root only when it is a perfect square, and 87 is not.

Its decimal begins $9.32737905\ldots$ and simply keeps going, with no block of digits ever settling into a repeat.

Can the Square Root of 87 Be Simplified?

No. $\sqrt{87}$ is already in simplest radical form, and here is the reason, shown the same way you would when simplifying radical expressions.

Prime factorization check.

$87 = 3 \times 29$

Both 3 and 29 are prime.

Neither factor is repeated, so there is no perfect-square factor to pull out.

$\sqrt{87} = \sqrt{3 \times 29}$

Because nothing pairs up, $\sqrt{87}$ stays exactly as it is. Contrast this with a nearby value like the square root of 85, which also stays under the radical because $85 = 5 \times 17$ has no repeated prime either.

How Do You Find the Square Root of 87 by Long Division?

Long division builds the decimal one digit at a time. Here is the value to two decimals, one step per line.

Pair the digits from the decimal point: $\overline{87}.\overline{00},\overline{00}$

Largest square $\leq 87$ is $9^2 = 81$, so the first digit is 9, remainder 6.

Bring down 00: the value is 600; double the quotient (9) to get 18, and $183 \times 3 = 549 \leq 600$, so the next digit is 3, remainder 51.

Bring down 00: the value is 5100; double 93 to get 186, and $1862 \times 2 = 3724 \leq 5100$, so the next digit is 2, remainder 1376.

That gives $\sqrt{87} \approx 9.32$, and continuing yields $9.3274$. The digits never repeat, confirming the number is irrational.

Examples Of the Square Root of 87

Example 1

Show that $\sqrt{87}$ is in simplest radical form.

$87 = 3 \times 29$

Both primes appear once, so no square factor exists.

Final answer: $\sqrt{87}$ is already simplest form.

Example 2

A student writes $\sqrt{87} = \sqrt{81} + \sqrt{6} = 9 + \sqrt{6}$. Is that correct?

The tempting move is to split 87 into $81 + 6$ and take the root of each piece. Test it: $9 + \sqrt{6} \approx 9 + 2.449 = 11.449$, but $\sqrt{87} \approx 9.33$. They do not match, so the step is wrong.

A square root does not distribute over a sum: $\sqrt{a + b} \neq \sqrt{a} + \sqrt{b}$. There is no shortcut here; $\sqrt{87}$ is simply $\sqrt{87}$.

Final answer: $\sqrt{87} \approx 9.3274$

Example 3

Evaluate $(\sqrt{87})^2$.

$(\sqrt{87})^2 = 87$

Final answer: $87$

Example 4

Estimate $\sqrt{87}$ without a calculator.

$9^2 = 81$ and $10^2 = 100$, so $\sqrt{87}$ is between 9 and 10.

87 is much closer to 81 than to 100, so the estimate lands near $9.3$.

Final answer: about $9.3$ (actual $\approx 9.3274$).

Example 5

Simplify $\sqrt{87} \times \sqrt{3}$.

$\sqrt{87} \times \sqrt{3} = \sqrt{87 \times 3}$

$\sqrt{87} \times \sqrt{3} = \sqrt{261} = \sqrt{9 \times 29} = 3\sqrt{29} \approx 16.155$

Final answer: $3\sqrt{29}$. Multiplying by $\sqrt{3}$ introduces a pair of 3s under the radical, which is what finally lets a factor step out.

Common Mistakes

Mistake 1: Splitting the root over addition

Where it slips in: Rewriting $\sqrt{87}$ as $\sqrt{81} + \sqrt{6}$.

Don't do this: Writing $\sqrt{87} = 9 + \sqrt{6}$ and simplifying from there.

The correct way: A root splits over multiplication, not addition. Since $87 = 3 \times 29$ has no repeated prime, $\sqrt{87}$ does not simplify at all. Assuming every root must reduce is the most common first-instinct error with numbers like 87.

Mistake 2: Forcing a simplification that isn't there

Where it slips in: Expecting $\sqrt{87}$ to become "$a\sqrt{b}$" like its neighbours.

Don't do this: Writing $\sqrt{87} = 3\sqrt{29}$ by pulling a 3 out incorrectly.

The correct way: A factor only steps out when it appears twice. Here 3 and 29 each appear once, so nothing comes out; $3\sqrt{29}$ would square to $261$, not 87.

Mistake 3: Rounding too early

Where it slips in: Multi-step problems that reuse $\sqrt{87}$.

Don't do this: Replacing $\sqrt{87}$ with $9.3274$ at the start of a long calculation.

The correct way: Keep $\sqrt{87}$ exact until the final step, then round once. Early rounding lets error grow through every operation that follows.

Conclusion

The square root of 87 is about $9.3274$, it is irrational, and it stays as $\sqrt{87}$ because $87 = 3 \times 29$ hides no perfect-square factor. Confirm the simplest form by factoring, estimate with the nearest perfect squares, and keep the exact radical until the last step. To strengthen these radical skills with a teacher, explore Bhanzu's algebra tutor, work with a high school math tutor, or join structured math tutoring.

Want to practice with a guide? Book a free demo class and work through radical problems step by step.

Read More

• Square Root 1 to 30, the full reference table of roots from 1 to 30.

• Square root tricks, quick ways to estimate roots that will not simplify.

• Square Root of 78, a nearby root worth comparing.

• Square Root of 68, a root that does simplify, unlike 87.

• Square Root of 65, another value close to a whole number.