Question

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Determine the two closest integers for each irrational number.
the square root of 19
the square root of 50
the square root of 117

Answer 1

Answers:

• a) 4 and 5
• b) 7 and 8
• c) 10 and 11

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Work Shown:

List out the perfect squares

{1,4,9,16,25,36,49,64,81,100,121}

We stop once we reach 117 or just a bit higher.

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Then notice that

`16 < 19 < 25 \ ... \ \text{19 between the squares 16 and 25}\\\\√(16) < √(19) < √(25) \ ... \ \text{ apply square root to all sides}\\\\4 < √(19) < 5`

Which shows
`√(19)` is between 4 and 5. Therefore, the two closest integers to
`√(19)` are 4 and 5.

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We follow the same steps for
`√(50)`

So,

`49 < 50 < 64\\\\√(49) < √(50) < √(64)\\\\7 < √(50) < 8`

So the square root of 50 is between 7 and 8.

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And finally,

`100 < 117 < 121\\\\√(100) < √(117) < √(121)\\\\10 < √(117) < 11`

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Using a calculator, we find that

`√(19) \approx 4.36`

`√(50) \approx 7.07`

`√(117) \approx 10.82`

which helps confirm our answers.