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Square Root of 115 — Value, Simplification, and Steps

Algebra

TL;DR

The square root of 115 is approximately 10.724, and it is irrational because 115=5×23 has no perfect-square factor. This article gives the value, shows why ( \sqrt{115} ) is already in simplest radical form, and works the long-division method step by step.

The Value of the Square Root of 115

The square root of 115 is approximately ( \sqrt{115} \approx 10.724 ), an irrational number whose decimal never terminates or repeats. Because 115 is not a perfect square, no whole number squares to give it exactly.

Quick Answer:
Result: ( \sqrt{115} \approx 10.724 )
Notation: Radical form ( \sqrt{115} ); decimal ≈10.724
Method shown: Prime factorization to test for simplification, then long division
Rational or irrational: Irrational, since 115=5×23 has no perfect-square factor
Exact form: ( \sqrt{115} ) (cannot be simplified further)

Quick Reference Table

Where the Square Root of 115 Appears

( \sqrt{115} ) appears whenever a distance or diagonal lands on 115 under the root. In coordinate geometry, the distance between points that differ by legs summing to 115 in squares evaluates to ( \sqrt{115} ). It also appears in standard-deviation and root-mean-square calculations, where a variance of 115 gives a spread of approximately 10.724 units.

What the Square Root of 115 Means

The square root of a number is the value that, multiplied by itself, gives that number. So ( \sqrt{115} ) is the number ( x ) with ( x^2=115 ).

Since ( 10^2=100 ) and ( 11^2=121 ), the root lies between 10 and 11, closer to 11.

How to Compute the Square Root of 115

Method 1: Prime factorization (to test for simplification)

Break 115 into primes. ( 115=5\times23 ). Both 5 and 23 are prime, and neither is repeated. No factor appears twice, so no square can be pulled out. Result: ( \sqrt{115} ) is already in simplest radical form.

Method 2: Estimation between perfect squares

Find the nearest perfect squares. ( 10^2=100 ) and ( 11^2=121 ). So ( 10 < \sqrt{115} < 11 ). Since 115 is closer to 121 than to 100, the root leans toward 11, near 10.7. Estimate: ( \sqrt{115} \approx 10.7 ).

Method 3: Long division (for a precise decimal)

Pair the digits: Largest square ≤1 is ( 1^2=1 ); first digit is 1, remainder 0, bring down 15 to get 15. Double the quotient (1→2); find d with 2d×d≤15; adding 7 gives 144 which is acceptable. Continue this process to refine further until you estimate ( 115 \approx 10.724 ).

Common Mistakes With Square Root of 115

1. Mistake 1: Trying to simplify ( \sqrt{115} ) as if it has smaller radicals.
   Correct approach: ( \sqrt{115}=\sqrt{5}\sqrt{23} ) does not simplify further.

2. Mistake 2: Writing ( 115=10.724 ) with an equals sign.
   Correct approach: Use ( \sqrt{115} ) for exact answers.

3. Mistake 3: Estimating as "about 12."
   Correct approach: Recognize the nearest perfect squares.

Conclusion

The square root of 115 is approximately 10.724 and is irrational. ( 115=5×23 ), so ( \sqrt{115} ) is already in simplest radical form. It lies between 10 and 11, closer to 11.