Question Video: Approximating the Size of Irrational Numbers Mathematics • 8th Grade

Michael is trying to find which two whole numbers lie on either side of √51. He decides that it will be helpful to use what he has been taught about square numbers. What is the greatest square number below 51? What is the smallest square number above 51? Hence, determine the two consecutive whole numbers that √51 lies between.

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Video Transcript

Michael is trying to find which two whole numbers lie on either side of the square root of 51. He decides that it will be helpful to use what he has been taught about square numbers. What is the greatest square number below 51? What is the smallest square number above 51? Hence, determine the two consecutive whole numbers that the square root of 51 lies between.

First of all, we should think about what we know about square numbers. A square number is the result of multiplying an integer by itself. For example, one squared is one, and two squared is four. So, we call four a square number. It’s helpful to memorize the first 10 or so square numbers. Three squared is nine. When we’re listing out square numbers, the first three are one, four, nine, then four squared, which is 16, five squared, which is 25, six squared, which is 36, seven squared, which is 49, eight squared, which is 64, then 81, and then 100. These are the first 10 square numbers, and we can use that to answer the questions below.

What is the greatest square number below 51? This means we’re looking for the square number that’s closest to 51 but not over. And that’s gonna be 49. The next question is similar. What is the smallest square number above 51? That means we need the square number that’s closest to 51 but must be greater than 51. And here, that’s going to be 64. If we put 49 and 64 on a number line, 51 would fall between them but closer to 49.

Now, on another number line, we could put the square root of 49 and the square root of 64. Again, when we add the square root of 51, it falls closer to the square root of 49 than it does to the square root of 64. But we know that the square root of 49 is seven, and the square root of 64 is eight. Which means we can say that the square root of 51 will fall somewhere between the consecutive whole numbers of seven and eight.

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