Let’s solve each problem step by step. We’re estimating squares and cubes of numbers — some are easy, others we can round or use nearby whole numbers to estimate.
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a) 2.5²
2.5 × 2.5
We know: 2 × 2 = 4, and 3 × 3 = 9 → so it’s between 4 and 9.
Actually, 2.5 is halfway between 2 and 3.
(2 + 0.5)² = 2² + 2×2×0.5 + 0.5² = 4 + 2 + 0.25 =
6.25
✔ Exact value:
6.25
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b) 2.8²
Close to 3² = 9, but a bit less.
2.8 = 3 - 0.2
So (3 - 0.2)² = 9 - 2×3×0.2 + 0.04 = 9 - 1.2 + 0.04 =
7.84
✔ Exact value:
7.84
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c) 5.5²
Halfway between 5 and 6.
5.5 × 5.5 = (5 + 0.5)² = 25 + 2×5×0.5 + 0.25 = 25 + 5 + 0.25 =
30.25
✔ Exact value:
30.25
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d) 5.1²
Close to 5² = 25
5.1 = 5 + 0.1
(5 + 0.1)² = 25 + 2×5×0.1 + 0.01 = 25 + 1 + 0.01 =
26.01
✔ Exact value:
26.01
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e) 1.9³
Close to 2³ = 8
1.9 = 2 - 0.1
Use approximation: (2 - 0.1)³ ≈ 8 - 3×4×0.1 + 3×2×0.01 - 0.001
Wait — better to compute directly:
1.9 × 1.9 = 3.61
Then 3.61 × 1.9
Break it down:
3.61 × 2 = 7.22
Minus 3.61 × 0.1 = 0.361
→ 7.22 - 0.361 =
6.859
✔ Exact value:
6.859
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f) 5.01³
Very close to 5³ = 125
5.01 = 5 + 0.01
Approximate using binomial:
(5 + 0.01)³ ≈ 125 + 3×25×0.01 + 3×5×0.0001 + ...
= 125 + 0.75 + 0.0015 + tiny bit ≈
125.7515
But let’s calculate more precisely:
First, 5.01 × 5.01 = ?
5 × 5 = 25
5 × 0.01 = 0.05 → twice that = 0.10
0.01 × 0.01 = 0.0001
→ 25 + 0.10 + 0.0001 =
25.1001
Now multiply by 5.01:
25.1001 × 5 = 125.5005
25.1001 × 0.01 = 0.251001
Add them: 125.5005 + 0.251001 =
125.751501
✔ Exact value:
≈125.75 (to 2 decimal places)
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g) 9.97³
Close to 10³ = 1000
9.97 = 10 - 0.03
Use approximation: (10 - 0.03)³ ≈ 1000 - 3×100×0.03 + 3×10×0.0009 - 0.000027
= 1000 - 9 + 0.027 - 0.000027 ≈
991.026973
Or compute step by step:
First, 9.97 × 9.97
= (10 - 0.03)² = 100 - 0.6 + 0.0009 =
99.4009
Now multiply by 9.97:
99.4009 × 10 = 994.009
Minus 99.4009 × 0.03 = 2.982027
→ 994.009 - 2.982027 =
991.026973
✔ Exact value:
≈991.03 (to 2 decimal places)
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h) 0.504²
Close to 0.5² = 0.25
0.504 = 0.5 + 0.004
(0.5 + 0.004)² = 0.25 + 2×0.5×0.004 + (0.004)²
= 0.25 + 0.004 + 0.000016 =
0.254016
✔ Exact value:
≈0.254 (to 3 decimal places)
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Final Answer:
a) 6.25
b) 7.84
c) 30.25
d) 26.01
e) 6.859
f) 125.75
g) 991.03
h) 0.254
Parent Tip: Review the logic above to help your child master the concept of power and roots worksheet.
