What Is A Square Root?
The square root of a number $n$ is the value $r$ for which $r^2 = n$. So the square root of 93 is the number that, multiplied by itself, gives 93.
No whole number does that. $9^2 = 81$ is too small, and $10^2 = 100$ is too big, so $\sqrt{93}$ falls between 9 and 10. Every positive number has two square roots, one positive and one negative, but the symbol $\sqrt{93}$ always means the positive, or principal, root.
Where Does √93 Appear?
$\sqrt{93}$ is the space diagonal of a rectangular box with edges 2, 5, and 8. The 3D version of the Pythagorean theorem gives the diagonal as $\sqrt{2^2 + 5^2 + 8^2} = \sqrt{4 + 25 + 64} = \sqrt{93}$. Because 93 sits between the perfect squares $81 = 9^2$ and $100 = 10^2$, the value lands between 9 and 10, closer to 10. Anywhere a length is built from squared parts that total 93, this irrational number is the exact answer.
Quick Reference Table
Number $n$ | $\sqrt{n}$ (approx.) | Rational or Irrational |
|---|---|---|
81 | 9 | Rational |
85 | 9.2195 | Irrational |
90 | 9.4868 | Irrational |
93 | 9.6437 | Irrational |
96 | 9.7980 | Irrational |
100 | 10 | Rational |
121 | 11 | Rational |
Is The Square Root Of 93 Rational Or Irrational?
$\sqrt{93}$ is an irrational number. It cannot be written as a fraction $\frac{p}{q}$ of two integers, and its decimal runs on forever without repeating.
The quick reason comes from the factors. A whole number has a rational square root only when it is a perfect square. Writing 93 as a product of primes gives:
$$93 = 3 \times 31$$
Both 3 and 31 appear to the first power, so there is no repeated prime to pull out as a pair, and 93 is not a perfect square. That forces $\sqrt{93}$ to be irrational. The same style of argument shows every non-perfect-square whole number has an irrational root, the way you can prove that root 3 is irrational or prove that root 7 is irrational by contradiction.
How Do You Find √93? (Long Division Method)
Prime factorization tells you $\sqrt{93}$ cannot be simplified, but it does not give the decimal. For that, use long division. Here is the method, one step per line.
Step 1: Write 93 as $\overline{93}.\overline{00},\overline{00},\overline{00}$, pairing digits from the decimal point.
Step 2: Find the largest number whose square is at most 93. $9^2 = 81 \leq 93$ Write 9 as the first digit of the answer.
Step 3: Subtract, then bring down the next pair. $93 - 81 = 12$ Bring down $00$ to get $1200$.
Step 4: Double the current answer, 9, to get 18. Find a digit $d$ with $(180 + d) \times d \leq 1200$. $186 \times 6 = 1116 \leq 1200$ The answer is now $9.6$.
Step 5: Subtract and bring down again. $1200 - 1116 = 84$ Bring down $00$ to get $8400$.
Step 6: Double 96 to get 192. Find $d$ with $(1920 + d) \times d \leq 8400$. $1924 \times 4 = 7696 \leq 8400$ The answer is now $9.64$.
Step 7: Continue once more. $8400 - 7696 = 704$ Bring down $00$ to get $70400$ $19283 \times 3 = 57849 \leq 70400$ The answer is now $9.643$.
Carrying the process one more place gives $\sqrt{93} \approx 9.6437$. The division never ends, which is exactly what an irrational value looks like on paper.
Examples Of √93
Example 1
Estimate $\sqrt{93}$ to the nearest whole number.
$9^2 = 81$ $10^2 = 100$ 93 is between 81 and 100, so the root is between 9 and 10. Since 93 is closer to 100, round up. Final answer: 10.
Example 2
A student simplifies $\sqrt{93}$ as $\sqrt{9} \times \sqrt{10.33}$ and stops. What went wrong?
The wrong path first. Splitting 93 as $9 \times 10.33$ does give $\sqrt{9} \times \sqrt{10.33} = 3\sqrt{10.33}$, but $10.33$ is not a whole number, so nothing was actually simplified. To simplify a radical you factor out a perfect square that divides the number evenly. $93 = 3 \times 31$ Neither factor is a perfect square, and no perfect square other than 1 divides 93. Final answer: $\sqrt{93}$ is already in simplest form.
Example 3
Evaluate $(\sqrt{93})^2$.
By definition, squaring undoes a square root. $(\sqrt{93})^2 = 93$ Final answer: 93.
Example 4
Find the value of $2\sqrt{93}$ to two decimal places.
$\sqrt{93} \approx 9.6437$ $2 \times 9.6437 = 19.2874$ Round to two places. Final answer: $19.29$.
Example 5
A cube-shaped tank has a space diagonal of $\sqrt{93}$ metres from combined edges 2, 5, and 8. Confirm the diagonal.
$2^2 + 5^2 + 8^2 = 4 + 25 + 64$ $= 93$ $\text{diagonal} = \sqrt{93} \approx 9.64 \text{ m}$ Final answer: about $9.64$ metres.
Common Mistakes
Mistake 1: Treating 93 as though it could be simplified
Where it slips in: A learner assumes every radical breaks down into something smaller.
Don't do this: Writing $\sqrt{93} = \sqrt{3} \times \sqrt{31}$ and calling it simpler.
The correct way: That split is valid but not simpler, since neither 3 nor 31 is a perfect square. Most students who reach for a factor pair forget to check that one factor is a perfect square. $\sqrt{93}$ stays $\sqrt{93}$.
Mistake 2: Rounding too early
Where it slips in: Multi-step problems where $\sqrt{93}$ appears in the middle.
Don't do this: Replacing $\sqrt{93}$ with $9.64$ at the start and carrying that value through every later step.
The correct way: Keep $\sqrt{93}$ in radical form until the final step, then round once. Early rounding compounds error across multiplications.
Mistake 3: Forgetting the negative root when solving an equation
Where it slips in: Solving $x^2 = 93$.
Don't do this: Writing only $x = \sqrt{93}$.
The correct way: The equation $x^2 = 93$ has two solutions, $x = \sqrt{93}$ and $x = -\sqrt{93}$, written together as $x = \pm\sqrt{93}$. The bare symbol $\sqrt{93}$ still means the positive root alone.
Conclusion
The square root of 93 is an irrational number, about $9.6437$, and $\sqrt{93}$ is already its simplest form because $93 = 3 \times 31$ carries no perfect-square factor. Long division gives the decimal to any precision you need, while prime factorization confirms the radical will not reduce. To take radicals further with a teacher, explore Bhanzu's algebra tutor sessions or structured math classes online. You can also book a free demo class to see the approach in action.
Read More
Square root tricks for estimating roots quickly
Simplifying radical expressions for the full reduction method
Square root 1 to 30 as a reference chart
Square root of 85, a nearby irrational value
Square root of 137, another non-simplifiable root
Squares and square roots for the underlying concept
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