Square Root of 10 - Value, Calculation, Examples
Square Root of 10 - Value, Calculation, Examples
TL;DR
The square root of 10 (10\sqrt{10}10) is approximately 3.16227766. This article gives the exact form, the decimal approximation to several places, the long-division method to compute it by hand, where 10\sqrt{10}10 shows up in geometry (the diagonal of a 1×3 rectangle), and why it's irrational.
The square root of 10 is approximately 3.16227766, an irrational number whose decimal expansion never terminates or repeats.
Quick Answer:
- Result: 10≈3.16227766\sqrt{10} \approx 3.16227766
- Notation: 10\sqrt{10}10 (exact, irrational), or 3.162 to three decimal places
- Method shown: Long division
- Exact form: 10\sqrt{10}10 — cannot be simplified further (10 has no perfect-square factors greater than 1)
- Irrational? Yes — non-terminating, non-repeating decimal
Quick Reference Table
| Number nnn | n\sqrt{n}n (approx.) | Rational or Irrational |
|---|---|---|
| 1 | 1 | Rational |
| 4 | 2 | Rational |
| 9 | 3 | Rational |
| 10 | 3.162 | Irrational |
| 12 | 3.464 | Irrational |
| 16 | 4 | Rational |
| 25 | 5 | Rational |
| 49 | 7 | Rational |
| 64 | 8 | Rational |
| 100 | 10 | Rational |
Where 10\sqrt{10}10 Appears
10\sqrt{10}10 shows up as the diagonal of a 1×3 rectangle — the Pythagorean theorem gives 12 + 32 = 10\sqrt{1^2 + 3^2} = \sqrt{10}. It also appears in engineering as the length of a unit step in base-10 logarithmic scales — a power ratio of 10 dB corresponds to a voltage ratio of 10\sqrt{10}10.
What Is a Square Root?
The square root of a number nnn is a value rrr such that r2 = n. The square root of 10 is the number that, multiplied by itself, gives 10.
There is no integer that does this. So 10\sqrt{10}10 lies between 3 and 4, somewhere closer to 3.
Because 10 is not a perfect square, 10\sqrt{10}10 is an irrational number — its decimal goes on forever without ending or repeating. The first 12 digits are: 10 = 3.16227766017…\sqrt{10} = 3.16227766017…
Is the Square Root of 10 Rational or Irrational?
10\sqrt{10}10 is irrational — meaning it cannot be written as a fraction pq of two integers, and its decimal expansion neither terminates nor repeats.
How to Find 10\sqrt{10}10 — Long Division Method
Step 1: Pair the digits from the decimal point. 10.00‾00‾00‾…
Step 2: Find the greatest number whose square is ≤10. That's 3. So, the first digit of the quotient will be 3.
Step 3: Subtract: 10 − 9 = 1. Bring down the next pair of zeros: 100.
Step 4: Double the current quotient: 3×2=6. Find a digit d such that (60+d)×d≤100. Try d=1: 61×1=61. Therefore, the quotient becomes 3.1.
Step 5: Subtract 61 from 100: remainder 39. Bring down next pair: 3900.
Step 6: Find d such that (620+d)×d≤3900. Try d=6: 626×6=3756. So, the quotient becomes 3.16.
Step 7: Continue this process to find further decimal places, ultimately yielding a longer decimal approximation.
Common Mistakes With Square-Root-of-10
Mistake 1: Reporting 10 = ±3.162
Where it slips in: Solving x2 = 10.
Mistake 2: Trying to simplify 10\sqrt{10}10
The correct way: 10\sqrt{10}10 is already in simplest radical form.
Mistake 3: Rounding too early
Keep 10\sqrt{10}10 as 10\sqrt{10}10 symbolically until the final step.
Frequently Asked Questions
What is the value of 10\sqrt{10}10?
10≈3.16227766.
Is 10\sqrt{10}10 rational or irrational?
Irrational.
What is the square of 10\sqrt{10}10?
(10)2=10(\sqrt{10})^2 = 10.
Can 10\sqrt{10}10 be simplified?
No, it is already in simplest form.
Where does 10\sqrt{10}10 appear in geometry?
As the diagonal of a 1×3 rectangle.