Expert: Richard J. Raridon - 2/25/2008

Question

Find the distance between the points.
(√7, -√3)  &  (-√10, √5)

The distance formula is shown below.

d=√ (x2 - x1)² + (y2 - y1)²

If we replace x2     by -√10, then we replace x1 by √7

If we replace  y2 by  √5 , then we replace y1 by -√3

Substitute and simplify.

  d= √(-√10 - √7)² + [√5 – (-√3) ]²
  d= √(-√10 - √7)² + (√5 + √3)²

Square each term and simplify the result. Recall that the product of two radicals is the radical of the product.

  d= √(-√10 - √7)(-√10 - √7) + (√5 + √3) (√5 + √3)
  d=√(10+2√10•7 +7)(5+2√5•3 +3)
  d=√(71 + 2• √70) + (8 + 2• √15)
  d=√(25+2 • √70 +2 •√15)

Evaluate d.

  d is approximately 7.034 (rounded to the nearest thousandth)

The distance between the given points is approximately 7.034

QUESTION: How did they round √(25+2 • √70 +2 •√15) to the nearest thousandth with out a calculator?  I understand all of the steps except this one.  


ANSWER: Well, I know how to take a square root with pencil and paper since we had to do that before calculators.

---------- FOLLOW-UP ----------

QUESTION: I didn't mean it like that.  My instructor does not allow the use of calculators period.  Which is why I added that particular detail.  The no calculator comment was to get you to give me as many details a possible for rounding, since I don't have a clue as to the real nature of those irrational numbers.

Answer
It's difficult to estimate square roots since they're non-linear.  For 70^1/2, for example, you know that 8^2 = 64 and 9^2 = 81 so you could make a guess at 8.2 and square 8.2 to check it.  for 15^1/2, that's a little less than 4 since 4^2 = 16, so guess 3.9
There's no way you could guess it to the nearest thousandth without a lot of multiplication.  
Back in the days when we only had slide rules we had to estimate the answer to know where to put the decimal point.  That's part of the problem with calculators.  You could hit a wrong button and never know that you got the wrong answer if you didn't have any idea of approximately what the answer should be.