Final answer:
To demonstrate why √31 is between 5 and 6, we compare it to the known squares of 5 and 6. Since 31 is between 25 (5²) and 36 (6²), √31 must fall between 5 and 6.
Step-by-step explanation:
To show why √31 is between 5 and 6, we need to find the squares of 5 and 6, because the square root of a number is one of the two identical factors that produce that number when multiplied together. Squaring 5 gives us 5² = 25, and squaring 6 gives us 6² = 36. Since 31 is between 25 and 36, the square root of 31 must be between the square roots of these two numbers, which are 5 and 6 respectively.
Mathematically, if 5 < x < 6, then 5² < x² < 6². Since 5² = 25 and 6² = 36, and 25 < 31 < 36, we can conclude that 5 < √31 < 6.
It may help to visualize this with an example of hypotenuse calculation: √(9 blocks)² + (5 blocks)² = 10.3 blocks, which is between 9 and 11 blocks, reinforcing the concept that the square root of a sum of squares falls between the square roots of the individual terms.