sqrt(18) - sqrt(17)
How does this expression equal
1/(sqrt(18) + sqrt(17)) ?
How would you change it like that?
And why is this good to use when not using a calculator?
(a-b)(a+b) = a^2-b^2, so
(√18-√17) * (√18+√17)/(√18+√17) = (18-17)/√(18+√17)
A calculator has roundoff error. √18 is very close to √17. Not too close in this case, but √188888 - √188887 are very close, and (floating-point) precision will be lost when trying to include the result with other numbers not so small.
If you add them, the roundoff occurs farther out, making it less important.
To explain how the expression √(18) - √(17) can be rewritten as 1/(√(18) + √(17)), we need to rationalize the denominator by multiplying both the numerator and denominator by the conjugate of the denominator, which is √(18) - √(17).
Multiplying the numerator and denominator by √(18) - √(17) gives us:
[√(18) - √(17)] * [√(18) - √(17)]
= (√(18) * √(18)) - (√(18) * √(17)) - (√(17) * √(18)) + (√(17) * √(17))
= (18 - √(306) - √(306) + 17)
= (18 + 17 - 2√(306))
= 35 - 2√(306)
So, now we have (35 - 2√(306)) in the numerator.
In the denominator, we have (√(18) + √(17)).
To express the reciprocal, we need to invert this fraction, which gives us 1/(√(18) + √(17)).
Therefore, the expression √(18) - √(17) is equivalent to 1/(√(18) + √(17)) after rationalizing the denominator.
This process is useful when not using a calculator because it allows us to simplify radical expressions and manipulate them algebraically, which can lead to more manageable forms that are easier to work with.