Describe how you would estimate the square root of a number that is not a perfect square with out using a calculator.
IF anyone could help me to have an understanding of what this question wants or means I'd appreciate it. Thanks
given a^2
guess x for sqrt a
a^2 = (x + y)^2
a^2 = x^2 + 2 x y + y^2
if my guess was perfect, y = 0
if just a little off, then y^2 is much less than 2 x y
so I approximate
2 x y = a^2 - x^2
for example if I want the square root of 10
a^2 = 10
guess x = 3 so x^2 = 9
2 (3) y = 10 - 9 = 1
y = 1/6
so my next guess at sqrt 10 is 3+1/6 or 3.167 = new x
well how far off is that?
3.167^2 = 10.03 , not bad. The next try using x = 3.167 will be about as close as your calulator could come:)
(my calculator says sqrt 10 = 3.16227766 )
Thanks I understand it much better now.
Estimating the square root of a nonperfect square without using a calculator involves finding an approximate value using various methods. Here's one approach:
1. Prime factorization method:
- Start with the given number and try to factor it into prime numbers.
- For example, if the number is 35, it can be factored as 5 * 7.
- Take the square root of each prime factor: √5 * √7.
- Since 5 is a prime number (not a perfect square), you can estimate its square root as approximately 2.24.
- For 7, its square root is approximately 2.65.
- Multiply the approximate square roots: 2.24 * 2.65 ≈ 5.94.
- Therefore, the estimation for √35 is approximately 5.94.
2. Repeated subtraction method:
- Start by assuming a whole number as the root, let's say 5.
- Subtract the perfect square less than or equal to the given number.
- If the result is negative, subtract the next smaller square instead.
- Repeat this process until you get a positive result, then start with the next digit of the root.
- For example, if the number is 67, start with 5 as the approximate root: √67 ≈ 5.
- Subtract the largest perfect square less than or equal to 67, which is 49: 67 - 49 = 18.
- Now, take the next digit of the root, which is 2: √67 ≈ 5.2.
- Subtract the square of 5.2 from 18: 18 - (5.2 * 5.2) = approximately 1.76.
- Repeat the process for the next digit of the root until you have the desired precision.
These methods provide approximate values for the square root of nonperfect square numbers. Remember that with these estimations, the accuracy increases with each additional digit used in the root approximation.