Video: Approximating the Size of Irrational Numbers

Which of the following statements is true? [A] If you evaluate √17, the answer is between 3 and 3.5. [B] If you evaluate √17, the answer is between 3.5 and 4. [C] If you evaluate √17, the answer is between 4 and 4.5. [D] If you evaluate √17, the answer is between 4.5 and 5. [E] If you evaluate √17, the answer is between 5 and 5.5.

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Video Transcript

Which of the following statements is true? All five of the statements want us to evaluate the square root of 17. Option (A) says, the answer is between three and 3.5. Option (B) the answer is between 3.5 and four. Option (C) the answer is between four and 4.5. Option (D) the answer is between 4.5 and five. Or option (E) the answer is between five and 5.5.

We will begin this question by drawing a number line to establish which two integer values the square root of 17 lies between. Let’s consider the integer values from three to six. We know that square rooting a number is the opposite of squaring it. As three squared is equal to nine, the square root of nine is equal to three. This means that we can put the radicals, the square root of nine, square root of 16, square root of 25, and square root of 36, on our number line.

The number 17 lies between 16 and 25. Therefore, the square root of 17 is greater than the square root of 16 and less than the square root of 25. We know that the square root of 16 is equal to four and the square root of 25 is equal to five. We can, therefore, conclude that the radical, square root of 17, lies between four and five. We can, therefore, eliminate options (A), (B), and (E).

4.5 is halfway between four and five. 17 is much closer to 16 than 25. It is one away from 16 but eight away from 25. This means that the square root of 17 will be closer to the square root of 16 than the square root of 25. We can, therefore, conclude that the square root of 17 lies between four and 4.5. This means that statement (C) is true. If you evaluate the square root of 17, the answer is between four and 4.5.

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