Let’s solve each problem one by one. We’ll simplify radicals, combine like terms, and use exponent rules where needed.
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1. √48
Break 48 into factors:
48 = 16 × 3 → √(16×3) = √16 × √3 =
4√3
✔ Answer:
4√3
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2. √75
75 = 25 × 3 → √(25×3) = √25 × √3 =
5√3
✔ Answer:
5√3
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3. 6√9
√9 = 3 → 6 × 3 =
18
✔ Answer:
18
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4. -6√9
Same as above: √9 = 3 → -6 × 3 =
-18
✔ Answer:
-18
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5. √2 · √2
Multiply under one root: √(2×2) = √4 =
2
Or: √2 × √2 = (√2)² =
2
✔ Answer:
2
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6. √12 · √3
√(12×3) = √36 =
6
✔ Answer:
6
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7. √(-100)
Square root of a negative number is not real — it’s imaginary.
√(-100) = √(100 × -1) = √100 × √(-1) =
10i
But if this class hasn’t covered imaginary numbers yet, you might write “not real” or leave blank. Since the worksheet includes other basic problems, maybe they expect
10i.
Wait — looking at later problems (like #12: i⁶), they DO cover imaginary numbers! So we’ll go with
10i
✔ Answer:
10i
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8. i⁴
Powers of i cycle every 4:
i¹ = i
i² = -1
i³ = -i
i⁴ = 1 ← because i⁴ = (i²)² = (-1)² = 1
✔ Answer:
1
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9. √48 ← Same as #1!
Already solved:
4√3
✔ Answer:
4√3
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10. √2 · √8
√(2×8) = √16 =
4
✔ Answer:
4
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11. 5√4
√4 = 2 → 5 × 2 =
10
✔ Answer:
10
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12. i⁶
i⁶ = i⁴ × i² = 1 × (-1) =
-1
Or: since powers repeat every 4, 6 mod 4 = 2 → i⁶ = i² = -1
✔ Answer:
-1
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13. (√7)²
Squaring a square root cancels out: (√7)² =
7
✔ Answer:
7
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14. -5(√2)²
First, (√2)² = 2 → then multiply by -5: -5 × 2 =
-10
✔ Answer:
-10
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15. √2 · √8 ← Same as #10!
Already solved:
4
✔ Answer:
4
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16. 4i · 3i
Multiply coefficients: 4 × 3 = 12
Multiply i’s: i × i = i² = -1
So: 12 × (-1) =
-12
✔ Answer:
-12
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17. 3(2i)²
First, (2i)² = 4i² = 4(-1) = -4
Then multiply by 3: 3 × (-4) =
-12
✔ Answer:
-12
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18. (2 + 3i)(1 - i)
Use FOIL:
First: 2×1 = 2
Outer: 2×(-i) = -2i
Inner: 3i×1 = 3i
Last: 3i×(-i) = -3i² = -3(-1) = +3
Now add all:
2 - 2i + 3i + 3 = (2+3) + (-2i+3i) =
5 + i
✔ Answer:
5 + i
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19. -4·8i
Just multiply: -4 × 8 = -32 → so
-32i
✔ Answer:
-32i
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20. i⁵
i⁵ = i⁴ × i = 1 × i =
i
Or: 5 mod 4 = 1 → i⁵ = i¹ = i
✔ Answer:
i
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## Final Answer:
1. 4√3
2. 5√3
3. 18
4. -18
5. 2
6. 6
7. 10i
8. 1
9. 4√3
10. 4
11. 10
12. -1
13. 7
14. -10
15. 4
16. -12
17. -12
18. 5 + i
19. -32i
20. i
Parent Tip: Review the logic above to help your child master the concept of imaginary numbers homework.
