Math worksheet for approximating square roots, featuring eight problems requiring students to estimate values like √200, √136, and others to the nearest tenth.

Let's solve each of these square roots and approximate them to the nearest tenth (i.e., one decimal place). We’ll use estimation techniques based on perfect squares.

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1) √200

- Find perfect squares near 200:
- $ 14^2 = 196 $
- $ 15^2 = 225 $
- So, $ \sqrt{200} $ is between 14 and 15.
- Since 200 is closer to 196 than 225:
- Try $ 14.1^2 = 198.81 $
- $ 14.2^2 = 201.64 $
- 200 is between 198.81 and 201.64 → closer to 14.1
- But let’s check:
$ 14.1^2 = 198.81 $
$ 14.15^2 = ? $
$ 14.15^2 = (14 + 0.15)^2 = 196 + 2(14)(0.15) + (0.15)^2 = 196 + 4.2 + 0.0225 = 200.2225 $

So $ \sqrt{200} \approx 14.1 $ or $ 14.2 $?
Since $ 14.1^2 = 198.81 $, $ 14.15^2 = 200.22 $, and 200 is just below that, so it's slightly less than 14.15.

Try $ 14.14^2 = (14.1 + 0.04)^2 = 14.1^2 + 2(14.1)(0.04) + (0.04)^2 = 198.81 + 1.128 + 0.0016 = 199.9396 $

Very close!
$ 14.14^2 ≈ 199.94 $, $ 14.15^2 ≈ 200.22 $

So $ \sqrt{200} \approx 14.14 $ → rounded to nearest tenth: 14.1

✔ Answer: 14.1

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2) √136

- Perfect squares:
- $ 11^2 = 121 $
- $ 12^2 = 144 $
- 136 is between 121 and 144
- Try $ 11.6^2 = 134.56 $
- $ 11.7^2 = 136.89 $
- 136 is between 134.56 and 136.89
- Closer to 11.7?
- Difference: 136 – 134.56 = 1.44
- 136.89 – 136 = 0.89 → closer to 11.7
- Try $ 11.65^2 = (11.6 + 0.05)^2 = 134.56 + 2(11.6)(0.05) + 0.0025 = 134.56 + 1.16 + 0.0025 = 135.7225 $
- Still low
- $ 11.67^2 = ? $
- $ 11.67^2 = (11.7 - 0.03)^2 = 136.89 - 2(11.7)(0.03) + 0.0009 = 136.89 - 0.702 + 0.0009 = 136.1889 $
- $ 11.66^2 = 11.67^2 - 2(11.67)(0.01) + (0.01)^2 ≈ 136.1889 - 0.2334 + 0.0001 ≈ 135.9556 $
- So 136 is between 11.66² and 11.67²
- Since 136 is closer to 136.1889 than to 135.9556 → closer to 11.67
- But we want nearest tenth, so compare 11.6 vs 11.7

We know:
- $ 11.6^2 = 134.56 $
- $ 11.7^2 = 136.89 $
- 136 is 1.44 above 134.56, and 0.89 below 136.89 → so closer to 11.7

So round to 11.7

✔ Answer: 11.7

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3) √7

- $ 2^2 = 4 $
- $ 3^2 = 9 $
- So √7 is between 2 and 3
- Try $ 2.6^2 = 6.76 $
- $ 2.7^2 = 7.29 $
- 7 is between 6.76 and 7.29
- 7 – 6.76 = 0.24
- 7.29 – 7 = 0.29 → closer to 2.6
- Try $ 2.64^2 = (2.6 + 0.04)^2 = 6.76 + 2(2.6)(0.04) + 0.0016 = 6.76 + 0.208 + 0.0016 = 6.9696 $
- $ 2.65^2 = 2.64^2 + 2(2.64)(0.01) + 0.0001 ≈ 6.9696 + 0.0528 + 0.0001 = 7.0225 $
- So √7 ≈ 2.645 → rounds to 2.6 (since 2.645 is closer to 2.6 than 2.7? Wait: 2.65 is midpoint)

Wait: To nearest tenth, look at hundredths digit.

2.645 → tenths digit is 6, hundredths is 4 → less than 5 → round down

So 2.6

But wait: 2.645 is actually halfway between 2.6 and 2.7? No — 2.65 is midpoint.

2.645 < 2.65 → so rounds to 2.6

✔ Answer: 2.6

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4) √74

- $ 8^2 = 64 $
- $ 9^2 = 81 $
- So √74 between 8 and 9
- Try $ 8.6^2 = 73.96 $
- $ 8.7^2 = 75.69 $
- 74 is very close to 73.96 → difference: 74 – 73.96 = 0.04
- 75.69 – 74 = 1.69 → much farther
- So √74 ≈ 8.602 → rounds to 8.6

✔ Answer: 8.6

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5) √116

- $ 10^2 = 100 $
- $ 11^2 = 121 $
- So √116 between 10 and 11
- Try $ 10.7^2 = 114.49 $
- $ 10.8^2 = 116.64 $
- 116 is between 114.49 and 116.64
- 116 – 114.49 = 1.51
- 116.64 – 116 = 0.64 → closer to 10.8
- So √116 ≈ 10.77 → rounds to 10.8

✔ Answer: 10.8

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6) √119

- $ 10.9^2 = 118.81 $
- $ 11.0^2 = 121 $
- 119 – 118.81 = 0.19
- 121 – 119 = 2 → so closer to 10.9
- Try $ 10.92^2 = (10.9 + 0.02)^2 = 118.81 + 2(10.9)(0.02) + 0.0004 = 118.81 + 0.436 + 0.0004 = 119.2464 $ → too high
- $ 10.91^2 = 118.81 + 2(10.9)(0.01) + 0.0001 = 118.81 + 0.218 + 0.0001 = 119.0281 $
- $ 10.90^2 = 118.81 $
- So √119 ≈ 10.905 → rounds to 10.9

✔ Answer: 10.9

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7) √188

- $ 13^2 = 169 $
- $ 14^2 = 196 $
- 188 is between 169 and 196
- $ 13.7^2 = 187.69 $
- $ 13.8^2 = 190.44 $
- 188 – 187.69 = 0.31
- 190.44 – 188 = 2.44 → closer to 13.7
- Try $ 13.72^2 = (13.7 + 0.02)^2 = 187.69 + 2(13.7)(0.02) + 0.0004 = 187.69 + 0.548 + 0.0004 = 188.2384 $ → too high
- $ 13.71^2 = 187.69 + 2(13.7)(0.01) + 0.0001 = 187.69 + 0.274 + 0.0001 = 187.9641 $
- $ 13.72^2 = 188.2384 $
- 188 is between 187.9641 and 188.2384 → closer to 13.71
- So √188 ≈ 13.71 → rounds to 13.7

✔ Answer: 13.7

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8) √12

- $ 3^2 = 9 $
- $ 4^2 = 16 $
- So √12 between 3 and 4
- $ 3.4^2 = 11.56 $
- $ 3.5^2 = 12.25 $
- 12 – 11.56 = 0.44
- 12.25 – 12 = 0.25 → closer to 3.5
- Try $ 3.46^2 = (3.4 + 0.06)^2 = 11.56 + 2(3.4)(0.06) + 0.0036 = 11.56 + 0.408 + 0.0036 = 11.9716 $
- $ 3.47^2 = 11.9716 + 2(3.46)(0.01) + 0.0001 ≈ 11.9716 + 0.0692 + 0.0001 = 12.0409 $
- So √12 ≈ 3.464 → rounds to 3.5

✔ Answer: 3.5

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✔ Final Answers:

1) √200 ≈ 14.1
2) √136 ≈ 11.7
3) √7 ≈ 2.6
4) √74 ≈ 8.6
5) √116 ≈ 10.8
6) √119 ≈ 10.9
7) √188 ≈ 13.7
8) √12 ≈ 3.5

Let me know if you'd like a step-by-step explanation for any specific one!