Square Root of 105 — Value and How to Find It
Square Root of 105 — Value and How to Find It
TL;DR
The square root of 105 is ( \sqrt{105} \approx 10.247 ), an irrational number that cannot be simplified because 105 = 3 × 5 × 7 has no repeated prime factor. This article shows the factorization check, the long-division method, where it appears, common mistakes, and worked examples.
Last updated on July 20, 2026
The square root of 105 is ( \sqrt{105} \approx 10.247 ), and it stays under the radical because 105 has no square factor.
Quick Answer:
Result: ( \sqrt{105} \approx 10.247 )
Notation: ( \sqrt{105} ) or ( 105^{1/2} )
Method shown: long division and prime-factorization check
Approximate value: 10.247 (to 3 decimal places, irrational)
Exact form: ( \sqrt{105} ) (already in simplest radical form)
Quick Reference Table
| Number | Simplified square root | Decimal (3 dp) |
|---|---|---|
| ( 100 ) | ( 10 ) | ( 10.000 ) |
| ( 104 ) | ( 2\sqrt{26} ) | ( 10.198 ) |
| ( 105 ) | ( \sqrt{105} ) | ( 10.247 ) |
| ( 106 ) | ( \sqrt{106} ) | ( 10.296 ) |
| ( 108 ) | ( 6\sqrt{3} ) | ( 10.392 ) |
| ( 121 ) | ( 11 ) | ( 11.000 ) |
Where the Square Root of 105 Appears
The square root of 105 is the side length of a square whose area is 105 square units. It also shows up as a diagonal: a rectangle with sides measuring roughly ( 5 ext{ and }21 ) has a diagonal of 5 + 100 style calculations, and any right triangle whose legs square-sum to 105 has a hypotenuse of exactly ( \sqrt{105} ).
What Is the Square Root of 105?
The square root of a number is the value that, multiplied by itself, gives that number. Since 105 is not a perfect square, it sits between ( 10^2 ) and ( 11^2 ), so its square root is between 10 and 11.
The result is an irrational number that never terminates and never repeats. Its prime factorization, 105 = 3 × 5 × 7, has no repeated factor, so nothing can leave the radical.
How to Find the Square Root of 105 (Methods)
Method 1: Prime-factorization check
Break 105 into prime factors.
105=3×5×7
A factor leaves a square root only when it appears twice. Every prime here appears once.
( \sqrt{105} = \sqrt{3 \times 5 \times 7} )
Final answer: ( \sqrt{105} ) does not simplify; it stays as ( \sqrt{105} ).
Method 2: Long division for the decimal value
Pair the digits of 105 from the right: 1|051. The largest square below 1 is 1, so the first digit is 1.
Bring down 05 to get 5, and find a digit ( x ) with ( 2x \times x \leq 500 ). Continuing the process gives the next decimals.
( 105 \approx 10.247 )
Final answer: ( 105 \approx 10.247 ) to three decimal places.
Common Mistakes With Square Root of 105
Mistake 1: Trying to pull a factor out
Correct way: Check the factorization first. Since 105 has no repeated prime, ( \sqrt{105} ) is already simplest.
Mistake 2: Rounding too early
Correct way: Carry the division to the required precision, ( \approx 10.247 ) before rounding.
Mistake 3: Confusing the square of 105 with its square root
Correct way: The square root asks for the number whose square is 105, which is ( \approx 10.247 ).
Frequently Asked Questions
What is the value of the square root of 105?
It is ( \sqrt{105} \approx 10.247 ), an irrational number.
Is the square root of 105 rational or irrational?
Irrational. Since 105 is not a perfect square, it is a non-terminating, non-repeating decimal.
Can ( \sqrt{105} ) be simplified?
No. Since 105 has no repeated prime factor, no whole number can leave the radical.
Is 105 a perfect square?
No. It lies between the perfect squares 100 and 121.
What is the negative square root of 105?
It is ( -\sqrt{105} \approx -10.247 ), since both ( +10.247 ) and ( -10.247 ) square to about 105.