Eighth Grade Square Roots of Fractions and Decimals Practice - Free Printable

Let’s solve each square root problem step by step. We’ll simplify when possible, and round to the nearest tenth if needed.

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Left Column (Problems 1–20):

1. √(1/4) = √1 / √4 = 1/2 → 0.5

2. √(4/9) = √4 / √9 = 2/3 ≈ 0.7 (rounded to nearest tenth)

3. √(9/16) = √9 / 16 = 3/4 → 0.75

4. √(16/25) = √16 / √25 = 4/5 → 0.8

5. √(25/36) = √25 / √36 = 5/6 ≈ 0.8 (rounded to nearest tenth)

6. √(3/4) = √3 / √4 ≈ 1.732 / 2 ≈ 0.9 (rounded to nearest tenth)

7. √(8/9) = √8 / √9 = (2√2)/3 ≈ (2×1.414)/3 ≈ 2.828/3 ≈ 0.9 (rounded to nearest tenth)

8. √(12/16) = √(3/4) = same as #6 → 0.9

9. √(18/25) = √18 / √25 = (3√2)/5 ≈ (3×1.414)/5 ≈ 4.242/5 ≈ 0.8 (rounded to nearest tenth)

10. √(2/9) = √2 / √9 ≈ 1.414 / 3 ≈ 0.5 (rounded to nearest tenth)

11. √0.9 → Let’s think: 0.9 is close to 1, but less. Try 0.95² = 0.9025 → too big. 0.94² = 0.8836 → too small. So between 0.94 and 0.95. Closer to 0.95? 0.948² = ? Actually, better to use calculator-style thinking: √0.9 ≈ 0.9 (since 0.95²=0.9025, so √0.9≈0.948… → rounds to 0.9)

Wait — let me recalculate more carefully:

Actually, √0.9 = √(9/10) = 3/√10 ≈ 3/3.162 ≈ 0.948 → rounds to 0.9? No! 0.948 rounded to nearest tenth is 0.9? Wait — tenths place is first decimal: 0.9 is 0.90, 0.948 is closer to 0.9 than to 1.0? No — 0.948 is closer to 0.9 or 1.0? Tenths: look at hundredths digit — 4, which is less than 5, so we round down → 0.9

But wait — actually, 0.948 is 0.9 when rounded to nearest tenth? Let me check:
Nearest tenth means one decimal place.
0.948 → look at second decimal (4), which is <5 → so round down → 0.9

BUT — that feels wrong because 0.95 would be halfway. Actually, standard rounding: 0.948 → 0.9 (yes). But let me confirm with actual value:
√0.9 ≈ 0.948683... → yes, rounds to 0.9

However, some might expect 0.95? No — rule is: look at next digit after the place you’re rounding to. For tenths, look at hundredths. Hundredths digit is 4 → round down → 0.9

But let me double-check problem 11: it says “round to the nearest tenth” — so 0.948... → 0.9

Wait — I think I made a mistake here. Let me calculate numerically:

√0.9 = ?

We know:
0.9^2 = 0.81
0.95^2 = 0.9025
0.94^2 = 0.8836
0.948^2 = (0.95 - 0.002)^2 ≈ 0.9025 - 2*0.95*0.002 + ... ≈ 0.9025 - 0.0038 = 0.8987 — too low
Better: try 0.9487^2 = ?

Actually, just accept: √0.9 ≈ 0.9487 → to nearest tenth: look at hundredths digit (4) → round down → 0.9

But this seems counterintuitive because 0.95 is closer to 1.0? No — 0.9487 is still closer to 0.9 than to 1.0? Distance to 0.9 is 0.0487, distance to 1.0 is 0.0513 — so actually closer to 0.9? Yes! So 0.9 is correct.

But let me verify with known values: many sources say √0.9 ≈ 0.9487 → rounded to nearest tenth is 0.9.

Okay, moving on.

12. √2.7 → Let’s estimate: 1.6^2 = 2.56, 1.7^2 = 2.89 → so between 1.6 and 1.7. 2.7 - 2.56 = 0.14, difference between 2.89-2.56=0.33 → so about 1.6 + 0.14/0.33 *0.1 ≈ 1.6 + 0.042 ≈ 1.642 → so approximately 1.64 → rounds to 1.6

Check: 1.64^2 = 2.6896, 1.65^2=2.7225 → 2.7 is closer to 2.6896? Difference: 2.7 - 2.6896 = 0.0104; 2.7225 - 2.7 = 0.0225 → so closer to 1.64 → so √2.7 ≈ 1.643 → rounds to 1.6

13. √9.2 → 3.0^2=9, 3.1^2=9.61 → so between 3.0 and 3.1. 9.2 - 9 = 0.2, total range 0.61 → so 3.0 + 0.2/0.61 *0.1 ≈ 3.0 + 0.0328 ≈ 3.0328 → check 3.03^2 = 9.1809, 3.04^2=9.2416 → 9.2 - 9.1809=0.0191, 9.2416-9.2=0.0416 → so closer to 3.03 → √9.2 ≈ 3.033 → rounds to 3.0

14. √7.8 → 2.8^2=7.84 → very close! 2.8^2=7.84, which is 0.04 over. 2.79^2 = (2.8 - 0.01)^2 = 7.84 - 2*2.8*0.01 + 0.0001 = 7.84 - 0.056 + 0.0001 = 7.7841 → 7.8 - 7.7841 = 0.0159, while 7.84 - 7.8 = 0.04 → so 2.79 is closer? Wait no: we want √7.8, so if 2.79^2=7.7841, 2.80^2=7.84 → 7.8 is closer to 7.7841? Difference 0.0159 vs 0.04 → yes, so √7.8 ≈ 2.793 → rounds to 2.8

Because 2.793 to nearest tenth: look at hundredths digit (9) ≥5 → round up → 2.8

Yes.

15. √0.17 → 0.4^2=0.16, 0.5^2=0.25 → so between 0.4 and 0.5. 0.17 - 0.16 = 0.01, range 0.09 → so 0.4 + 0.01/0.09 *0.1 ≈ 0.4 + 0.011 ≈ 0.411 → check 0.41^2=0.1681, 0.42^2=0.1764 → 0.17 - 0.1681=0.0019, 0.1764-0.17=0.0064 → so closer to 0.41 → √0.17 ≈ 0.4123 → rounds to 0.4

16. √0.35 → 0.5^2=0.25, 0.6^2=0.36 → very close to 0.6. 0.36 - 0.35 = 0.01, so √0.35 ≈ 0.5916 → rounds to 0.6

17. √32.97 → 5.7^2=32.49, 5.8^2=33.64 → 32.97 - 32.49 = 0.48, range 1.15 → so 5.7 + 0.48/1.15 *0.1 ≈ 5.7 + 0.0417 ≈ 5.7417 → check 5.74^2 = (5.7 + 0.04)^2 = 32.49 + 2*5.7*0.04 + 0.0016 = 32.49 + 0.456 + 0.0016 = 32.9476 → 32.97 - 32.9476 = 0.0224 → 5.75^2 = 33.0625 → too big. So √32.97 ≈ 5.742 → rounds to 5.7

18. √0.73 → 0.8^2=0.64, 0.9^2=0.81 → 0.73 - 0.64 = 0.09, range 0.17 → so 0.8 + 0.09/0.17 *0.1 ≈ 0.8 + 0.0529 ≈ 0.8529 → check 0.85^2=0.7225, 0.86^2=0.7396 → 0.73 - 0.7225=0.0075, 0.7396-0.73=0.0096 → so closer to 0.85 → √0.73 ≈ 0.8544 → rounds to 0.9? Wait — 0.8544 to nearest tenth: hundredths digit is 5 → round up → 0.9? But 0.85 is exactly halfway? Standard rule: if hundredths is 5 or more, round up. So 0.85 → 0.9? But 0.85 is 0.85, rounding to tenths: look at hundredths digit 5 → round up the tenths digit from 8 to 9 → so 0.9

But is that correct? 0.85 rounded to nearest tenth is 0.9? Yes, because 0.85 is exactly midway between 0.8 and 0.9, and convention is to round up when it's 5.

But our value is 0.8544, which is greater than 0.85, so definitely rounds to 0.9

19. √0.41 → 0.6^2=0.36, 0.7^2=0.49 → 0.41 - 0.36 = 0.05, range 0.13 → so 0.6 + 0.05/0.13 *0.1 ≈ 0.6 + 0.0385 ≈ 0.6385 → check 0.64^2=0.4096, 0.65^2=0.4225 → 0.41 - 0.4096=0.0004, very close! So √0.41 ≈ 0.6403 → rounds to 0.6

20. √0.05 → 0.2^2=0.04, 0.3^2=0.09 → 0.05 - 0.04 = 0.01, range 0.05 → so 0.2 + 0.01/0.05 *0.1 = 0.2 + 0.02 = 0.22 → check 0.22^2=0.0484, 0.23^2=0.0529 → 0.05 - 0.0484=0.0016, 0.0529-0.05=0.0029 → so closer to 0.22 → √0.05 ≈ 0.2236 → rounds to 0.2

Now Right Column (Problems 1–16):

1. √(1/4) = 1/2 → 0.5

2. √(9/16) = 3/4 → 0.75

3. √(100/16) = 10/4 = 2.5 → 2.5

4. √(25/100) = 5/10 = 0.5 → 0.5

5. √(25/225) = 5/15 = 1/3 ≈ 0.3 (rounded to nearest tenth)

6. √(4/100) = 2/10 = 0.2 → 0.2

11. √0.6 → 0.7^2=0.49, 0.8^2=0.64 → 0.6 - 0.49=0.11, range 0.15 → so 0.7 + 0.11/0.15 *0.1 ≈ 0.7 + 0.0733 ≈ 0.7733 → check 0.77^2=0.5929, 0.78^2=0.6084 → 0.6 - 0.5929=0.0071, 0.6084-0.6=0.0084 → so closer to 0.77 → √0.6 ≈ 0.7746 → rounds to 0.8 (hundredths digit 7≥5 → round up)

12. √3.7 → 1.9^2=3.61, 2.0^2=4.00 → 3.7 - 3.61=0.09, range 0.39 → so 1.9 + 0.09/0.39 *0.1 ≈ 1.9 + 0.023 ≈ 1.923 → check 1.92^2=3.6864, 1.93^2=3.7249 → 3.7 - 3.6864=0.0136, 3.7249-3.7=0.0249 → so closer to 1.92 → √3.7 ≈ 1.9235 → rounds to 1.9

13. √39.2 → 6.2^2=38.44, 6.3^2=39.69 → 39.2 - 38.44=0.76, range 1.25 → so 6.2 + 0.76/1.25 *0.1 ≈ 6.2 + 0.0608 ≈ 6.2608 → check 6.26^2=39.1876, 6.27^2=39.3129 → 39.2 - 39.1876=0.0124, 39.3129-39.2=0.1129 → so closer to 6.26 → √39.2 ≈ 6.261 → rounds to 6.3 (hundredths digit 6≥5 → round up)

14. √17.5 → 4.1^2=16.81, 4.2^2=17.64 → 17.5 - 16.81=0.69, range 0.83 → so 4.1 + 0.69/0.83 *0.1 ≈ 4.1 + 0.0831 ≈ 4.1831 → check 4.18^2=17.4724, 4.19^2=17.5561 → 17.5 - 17.4724=0.0276, 17.5561-17.5=0.0561 → so closer to 4.18 → √17.5 ≈ 4.1833 → rounds to 4.2 (hundredths digit 8≥5 → round up)

15. √0.37 → 0.6^2=0.36, 0.7^2=0.49 → 0.37 - 0.36=0.01, so very close to 0.6. 0.61^2=0.3721 → 0.3721 - 0.37=0.0021, while 0.37 - 0.36=0.01 → so 0.61 is closer? Wait: 0.6^2=0.36, 0.61^2=0.3721 → target 0.37 → difference to 0.36 is 0.01, to 0.3721 is 0.0021 → so closer to 0.61 → √0.37 ≈ 0.6083 → rounds to 0.6 (hundredths digit 0<5 → round down)

16. √9.75 → 3.1^2=9.61, 3.2^2=10.24 → 9.75 - 9.61=0.14, range 0.63 → so 3.1 + 0.14/0.63 *0.1 ≈ 3.1 + 0.0222 ≈ 3.1222 → check 3.12^2=9.7344, 3.13^2=9.7969 → 9.75 - 9.7344=0.0156, 9.7969-9.75=0.0469 → so closer to 3.12 → √9.75 ≈ 3.1225 → rounds to 3.1

Now let's compile all answers clearly.

Note: In left column, problems 1-10 are fractions, 11-20 are decimals. Right column has mixed.

I'll list them in order as per image.

Final Answers:

Left Column:
1. 0.5
2. 0.7
3. 0.75
4. 0.8
5. 0.8
6. 0.9
7. 0.9
8. 0.9
9. 0.8
10. 0.5
11. 0.9
12. 1.6
13. 3.0
14. 2.8
15. 0.4
16. 0.6
17. 5.7
18. 0.9
19. 0.6
20. 0.2

Right Column:
1. 0.5
2. 0.75
3. 2.5
4. 0.5
5. 0.3
6. 0.2
11. 0.8
12. 1.9
13. 6.3
14. 4.2
15. 0.6
16. 3.1

I think I have all. Let me verify a few critical ones again.

For left #11: √0.9 ≈ 0.9487 → to nearest tenth: 0.9 (correct)

Left #18: √0.73 ≈ 0.8544 → to nearest tenth: 0.9 (since 0.85 rounds up to 0.9)

Right #11: √0.6 ≈ 0.7746 → to nearest tenth: 0.8 (correct)

Right #13: √39.2 ≈ 6.261 → to nearest tenth: 6.3 (correct)

All seem consistent.

Final Answer:
Left Column Answers:
1. 0.5
2. 0.7
3. 0.75
4. 0.8
5. 0.8
6. 0.9
7. 0.9
8. 0.9
9. 0.8
10. 0.5
11. 0.9
12. 1.6
13. 3.0
14. 2.8
15. 0.4
16. 0.6
17. 5.7
18. 0.9
19. 0.6
20. 0.2

Right Column Answers:
1. 0.5
2. 0.75
3. 2.5
4. 0.5
5. 0.3
6. 0.2
11. 0.8
12. 1.9
13. 6.3
14. 4.2
15. 0.6
16. 3.1