What is `|1-sqrt2|+|2 - sqrt3| +|2sqrt2 - 3|`?

Expert Answers
txmedteach eNotes educator| Certified Educator

To simplify this expression, we need to get rid of the absolute value bars. To get rid of those, we need to determine whether the inside expressions are negative or positive. If they are negative, we simply multiply the inside by -1. If they are positive, we can simply remove the absolute value bars.

You can use a calculator to see which are negative or positive. Of course, if you are not allowed to use a calculator, knowledge of the approximate values of `sqrt2` and `sqrt3` are valuable in their own right for other math later on in your high school career, especially geometry! For example, right triangles and trigonometry use `sqrt2` and `sqrt3` all time!

Just for reference:

`sqrt2 ~~ 1.41`

`sqrt3 ~~ 1.73`

Now, let's go through the expressions one by one to see which are negative and which are positive:

`1-sqrt2 ~~ 1-1.41 = -0.41`

`2-sqrt3 ~~ 2 - 1.73 = 0.27`

`2sqrt2 - 3 ~~ 2(1.41)-3 = 2.82-3 = -0.18`

Thus, we see that the first and third expressions will require multiplying by -1, and we can leave the second expression alone. This analysis allows us to do the following:

`|1-sqrt2|+|2-sqrt3|+|2sqrt2-3| = (-1)(1-sqrt2)+2-sqrt3 + (-1)(2sqrt2-3)`

Let us now distribute those -1 terms to start evaluating the expression:


Now we can combine "like" terms, meaning we combine rational numbers with rational numbers and radicals with other radicals of the same kind:

`= (-1+2+3) + (sqrt2 - 2sqrt2) + (-sqrt3)`


`= 4 - sqrt2 - sqrt3`

Here, we have our final exact answer. If you want the approximate answer, you can convert the square roots to their approximate decimals:

`4-sqrt2-sqrt3~~4-1.41-1.73 = 0.86`