Square Root of 88 - Value, Simplified Form (2√22)

What Is A Square Root?

The square root of a number $n$ is the value $r$ for which $r^2 = n$, the number that, multiplied by itself, returns $n$. So $\sqrt{88}$ is the number whose square is 88.

No whole number fits: $9^2 = 81$ is just short and $10^2 = 100$ overshoots. That places $\sqrt{88}$ between 9 and 10, a little below the midpoint.

Where Does √88 Appear?

$\sqrt{88}$ is the natural answer to a distance question: a right triangle with legs of $2\sqrt{2}$ and $\sqrt{80}$ has a hypotenuse of $\sqrt{8 + 80} = \sqrt{88}$, and more simply, the diagonal of a rectangle with sides $2$ and $\sqrt{84}$ lands near the same place. It also appears in area work, where the side of a square that covers 88 square units measures $2\sqrt{22}$ units, a little under $9.39$.

Is 88 A Perfect Square?

No. A perfect square is an integer multiplied by itself, $1, 4, 9, 16, 25, \ldots, 81, 100$, and 88 is not on that list.

Because 88 is not a perfect square, its square root is not a whole number or a fraction. That is what makes $\sqrt{88}$ irrational.

Quick Reference Table

Is The Square Root Of 88 Rational Or Irrational?

$\sqrt{88}$ 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 88 is not.

Simplifying to $2\sqrt{22}$ does not change this. The leftover $\sqrt{22}$ is irrational, so the whole product stays irrational, which follows from the formal definition of a square root.

What Is The Square Root Of 88 In Simplest Radical Form?

The simplest radical form of $\sqrt{88}$ is $2\sqrt{22}$. For a square root you pull out factors that appear twice, the same grouping logic used in simplifying radical expressions.

Prime factorization method.

$88 = 2 \times 2 \times 2 \times 11$

$88 = 2^3 \times 11$

$\sqrt{88} = \sqrt{2^2 \times 2 \times 11}$

$\sqrt{88} = 2 \times \sqrt{2 \times 11}$

$\sqrt{88} = 2\sqrt{22}$

The three 2s hold one complete pair, which sends a single 2 outside the radical. The remaining $2 \times 11 = 22$ has no repeated factor, so it stays inside as $\sqrt{22}$. This is the same pattern as the square root of 8, which reduces to $2\sqrt{2}$.

How Do You Find √88 By Long Division?

Prime factorization gives the exact form; long division gives the decimal digit by digit. Here it is, one step per line, using the square root tricks that work without a calculator.

Pair the digits from the decimal point: $\overline{88}.\overline{00}\ \overline{00}$

The largest square not exceeding 88 is $81 = 9^2$, so the first digit is 9, remainder 7.

Bring down a pair of zeros to get 700; double the quotient to get 18, and find $d$ with $(180 + d)\times d \le 700$; $183 \times 3 = 549$, so the next digit is 3, quotient $9.3$.

Subtract to get remainder 151; bring down a pair of zeros to get 15100; double 93 to get 186, and find $d$ with $(1860 + d)\times d \le 15100$; $1868 \times 8 = 14944$, so the next digit is 8, quotient $9.38$.

Continue the same way and the digits settle at $\sqrt{88} \approx 9.3808$. The value never repeats, which is the signature of an irrational number. A quick sanity path: since 88 sits between $9^2 = 81$ and $10^2 = 100$, and closer to 81, the root should land just above 9, which it does.

Examples Of √88

Example 1

Simplify $\sqrt{88}$ to its radical form.

$88 = 2^3 \times 11$

$\sqrt{88} = 2 \times \sqrt{2 \times 11} = 2\sqrt{22}$

Final answer: $2\sqrt{22}$

Example 2

A student simplifies $\sqrt{88}$ and writes $\sqrt{88} = 4\sqrt{22}$ by treating the three 2s as two pairs. Where does it go wrong?

The tempting move is to imagine $2^3$ holds two pairs and send $2 \times 2 = 4$ outside. Test it: $4^2 \times 22 = 16 \times 22 = 352$, not 88, so 4 is too big.

Go back to the pairing rule. Three 2s make only one complete pair with a single 2 left over, so exactly one 2 leaves the radical. The answer is $2\sqrt{22}$, and the leftover 2 joins the 11 inside.

Final answer: $2\sqrt{22}$

Example 3

Evaluate $(\sqrt{88})^2$.

$(\sqrt{88})^2 = 88$

Final answer: $88$

Example 4

A square courtyard has an area of 88 square metres. Find its side length.

side $= \sqrt{88}$

side $= 2\sqrt{22} \approx 9.3808$

Final answer: $2\sqrt{22}$ m, about $9.38$ m.

Example 5

Simplify $\sqrt{88} + \sqrt{22}$.

$\sqrt{88} + \sqrt{22} = 2\sqrt{22} + \sqrt{22}$

$\sqrt{88} + \sqrt{22} = 3\sqrt{22} \approx 14.0712$

Final answer: $3\sqrt{22}$. Like radicals add the way like terms do; the shared $\sqrt{22}$ is what lets the sum collapse.

Common Mistakes

Mistake 1: Counting more pairs than the factorization holds

Where it slips in: Treating the three 2s in $2^3$ as two pairs.

Don't do this: Writing $\sqrt{88} = 4\sqrt{22}$.

The correct way: A square root releases a factor only for each complete pair. From $2^3$, there is one pair and one leftover 2, so a single 2 comes out and the answer is $2\sqrt{22}$.

Mistake 2: Splitting off the wrong factor

Where it slips in: Reaching for a factor pair that is not a perfect square.

Don't do this: Writing $\sqrt{88} = \sqrt{8}\times\sqrt{11}$ and calling it finished.

The correct way: $\sqrt{8}$ still simplifies to $2\sqrt{2}$, so the form is not yet simplest. Factor out the largest perfect square first: $88 = 4 \times 22$, giving $2\sqrt{22}$ in one clean step.

Mistake 3: Rounding too early

Where it slips in: Multi-step problems where $\sqrt{88}$ appears mid-calculation.

Don't do this: Replacing $\sqrt{88}$ with $9.38$ at the start and carrying that value through every step.

The correct way: Keep $2\sqrt{22}$ in exact form until the final line. Early rounding introduces error that compounds across multiplications and divisions.

Conclusion

The square root of 88 is $2\sqrt{22}$, roughly $9.3808$, and it is irrational because 88 has no whole-number square root. Write $88 = 2^3 \times 11$, pull out one factor for the single pair, and keep $2 \times 11 = 22$ inside. To build these radical and exponent skills with a teacher, explore Bhanzu's algebra tutor, get help with algebra, or join live math classes online.

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Read More

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

• Square Root of 50, a clean pairing example that reduces to $5\sqrt{2}$.

• Square Root of 85, a nearby non-perfect square with the factor 11's partner 17.

• Square Root of 20, another single-pair simplification down to $2\sqrt{5}$.

• Squares and Square Roots, how roots and powers connect across the number line.