Calculate the square root of 319

Problem

√(,319)

Solution

1. Check for perfect square factors by testing small prime numbers.

2. Test for divisibility by 2, 3, and 5. Since 319 is odd, does not end in 5 or 0, and the sum of its digits (3+1+9=13 is not divisible by 3, it is not divisible by these primes.

3. Test for divisibility by 7: 319÷7=45.57... (No).

4. Test for divisibility by 11: 319÷11=29

5. Identify the prime factorization of the radicand.

√(,319)=√(,11×29)

6. Conclude that since both 11 and 29 are prime numbers and neither is repeated, the square root cannot be simplified into a simpler radical form.

7. Estimate the decimal value by noting that 17=289 and 18=324 The value is slightly less than 18.

√(,319)≈17.86056...

Final Answer

√(,319)=√(,319)

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