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

$16 < 19 < 25 \ ... \ \text{19 between the squares 16 and 25}\\\\\sqrt{16} < \sqrt{19} < \sqrt{25} \ ... \ \text{ apply square root to all sides}\\\\4 < \sqrt{19} < 5$

Which shows $\sqrt{19}$ is between 4 and 5. Therefore, the two closest integers to $\sqrt{19}$ are 4 and 5.

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We follow the same steps for $\sqrt{50}$

So,

$49 < 50 < 64\\\\\sqrt{49} < \sqrt{50} < \sqrt{64}\\\\7 < \sqrt{50} < 8$

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

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And finally,

$100 < 117 < 121\\\\\sqrt{100} < \sqrt{117} < \sqrt{121}\\\\10 < \sqrt{117} < 11$

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Using a calculator, we find that

$\sqrt{19} \approx 4.36$

$\sqrt{50} \approx 7.07$

$\sqrt{117} \approx 10.82$

which helps confirm our answers.