Video: Finding Two Consecutive One-Decimal-Place Numbers That a Given Real Number Involving Roots Lies Between

Find the two consecutive one-decimal-place numbers that √151 lies between.

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Video Transcript

Find the two consecutive one-decimal-place numbers that the square root of 151 lies between.

This one decimal place is to the nearest tenth. And we want to look at the square root of 151. In solving problems like this, one strategy we often use is try and find a square number above and below the value we’re using. If we’re trying to find the square root of 151, we would want a square number both above and below this value. At first, you might think of 100, which is 10 squared. But when we think of 11 squared, that’s 121. And then 12 squared is 144. After that, we have 13 squared, which is 169. And that means we found the two values above and below 151.

When we go to put these values on a number line, we see that the square root of 151 is much closer to the square root of 144. This is because 151 is closer to 144 than it is to 169. We, of course, know that the square root of 144 is 12 and the square root of 169 is 13. And so, of course, we can say that the square root of 151 will be greater than 12 but less than 13. But sometimes, we want to be a bit more specific. To do that, imagine that we zoom in on this number line and we start with the square root of 144, which is 12. But then we want to look at the one-decimal-place numbers, which means we want 12.1, 12.2, and so on.

In order to decide where the square root of 151 will go, we calculate 12.1 squared, which is 146.41. If we jump over and look at 12.3 squared, we see that that is 151.29. It’s greater than 151. So, that means we found out that the square root of 151 cannot be more than 12.3. And that means we’ll need to check 12.2. 12.2 squared is 148.84, which is less than 151. And that means we found that the square root of 151 falls somewhere between 12.2 and 12.3. We can write this as the inequality 12.2 is less than the square root of 151 which is less than 12.3.

Before we finish, we just need to make a note about approximating higher roots. In all of our examples, we were approximating square roots. But cube roots and higher roots would follow the same procedure. For example, if we we’re dealing with cube numbers, you could list out the cube values. Make a number line. The cube root of one is one, the cube root of eight is two, the cube root of 27 is three. And that means if we were looking for the cube root of five, it would fall between one and two. And the same thing would work with the fourth root all the way up to the 𝑛th root.