Video: Arranging a Set of Irrational Numbers in the Form of Square Roots

Order the numbers on the cards from least to greatest. √137, 2√14, −3√11, √134

03:59

Video Transcript

Order the numbers on the cards from least to greatest.

From least to greatest means we’ll be moving from the smallest values to the largest values. Looking at these four card, I immediately spot that one of the four cards is a negative number. Because negative three times the square root of 11 is the only negative number in our set, we know that it will be the least value; it will be the smallest value we have. So I will just put that on a number line. We know that negative three times the square root of eleven will be less than zero.

Now we need some estimates for the next three cards. To estimate the square root of 137, I think about perfect squares that I know that are either slightly above or slightly below 137. For example, I know that 144 is a perfect square, and it’s pretty close to 137. If I take the square root of 144, I get 12. I also remembered that 121 is a perfect square; the square root of 121 equals 11. Because of this, I can estimate that the square root of 137 will fall somewhere between 11 and 12.

Now, let’s jump over to the square root of 134. The square root of 134 will also fall somewhere between the square root of 121 and 144, which means the square root of 134 is also between 11 and 12. However, we know that 134 is less than 137. That means that the square root of 134 will be less than the square root of 137. So we’ll graph them like this. The square root of 134 is going to come before the square root of 137 on a number line, and they’ll both fall between 11 and 12.

Finally, we need an estimate for two times the square root of 14. I think that the square root of 14 must be slightly less than the square root of 16. And the square root of 16 is four. I also know that the square root of nine is three, which means our square root of 14 will fall somewhere between three and four. But we can’t forget that in both of these cases, it would be two times the square root of 14. We can estimate that two times the square root of 14 will fall somewhere between six and eight.

We’ll plot on our number line that two times the square root of 14 falls somewhere between six and eight. Using our number line, we’ll move from left to right to list our cards in order from least to greatest. From least to greatest, our cards are negative three times the square root of 11, two times the square root of 14, the square root of 134, the square root of 137.

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