Let’s go through each problem one by one, simplifying carefully and checking our work.
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Simplify by factoring:
1) √28
→ 28 = 4 × 7 → √(4×7) = √4 × √7 =
2√7
2) √72
→ 72 = 36 × 2 → √(36×2) = √36 × √2 =
6√2
3) √32
→ 32 = 16 × 2 → √(16×2) = √16 × √2 =
4√2
4) √175
→ 175 = 25 × 7 → √(25×7) = √25 × √7 =
5√7
5) √108
→ 108 = 36 × 3 → √(36×3) = √36 × √3 =
6√3
6) √80
→ 80 = 16 × 5 → √(16×5) = √16 × √5 =
4√5
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Simplify by adding or subtracting:
7) 4√3 - √27
→ First simplify √27: √(9×3) = 3√3
→ So: 4√3 - 3√3 =
(4-3)√3 = 1√3 = √3
8) -10√11 - 11√11
→ Combine like terms: (-10 - 11)√11 =
-21√11
9) 3√5 + 7√5
→ (3+7)√5 =
10√5
10) -11√21 - 11√21
→ (-11 -11)√21 =
-22√21
11) -9√15 + 10√15
→ (-9 + 10)√15 =
1√15 = √15
12) 3√24 + 3√81
→ Simplify each radical:
√24 = √(4×6) = 2√6 → so 3×2√6 = 6√6
√81 = 9 → so 3×9 = 27
→ Final:
6√6 + 27
*(Note: These are not like terms — can’t combine further.)*
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Simplify by multiplying:
13) √10 · √14
→ Multiply under root: √(10×14) = √140
→ Factor 140: 140 = 4 × 35 → √(4×35) = 2√35
→ Answer:
2√35
14) ∛2 · ∛4
→ Same index? Yes, both cube roots.
→ ∛(2×4) = ∛8 =
2
15) 2√14 · 3√21
→ Multiply coefficients: 2×3 = 6
→ Multiply radicals: √14 × √21 = √(14×21) = √294
→ Factor 294: 294 = 49 × 6 → √(49×6) = 7√6
→ Total: 6 × 7√6 =
42√6
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Simplify by dividing and rationalizing the denominator:
16) ∛(10/9)
→ This is already simplified as a single radical. But if we want to write it as separate:
∛10 / ∛9 — but since 9 isn’t a perfect cube, and no common factors, this is simplest form.
→ However, sometimes they expect to rationalize even cube roots? Let’s check:
To rationalize ∛(10/9), multiply numerator and denominator inside the root by 3 to make denominator a perfect cube:
∛(10/9) = ∛(10×3)/(9×3) = ∛(30/27) = ∛30 / ∛27 = ∛30 / 3
✔ So final answer:
∛30 / 3
17) (1 + √2) / (3 + √5)
→ Rationalize denominator by multiplying numerator and denominator by conjugate of denominator: (3 - √5)
Numerator: (1 + √2)(3 - √5)
= 1·3 + 1·(-√5) + √2·3 + √2·(-√5)
= 3 - √5 + 3√2 - √10
Denominator: (3 + √5)(3 - √5) = 9 - 5 = 4
So result:
(3 - √5 + 3√2 - √10)/4
We can leave it like that — no like terms to combine.
18) 5 / (2 - √7)
→ Multiply numerator and denominator by conjugate: (2 + √7)
Numerator: 5(2 + √7) = 10 + 5√7
Denominator: (2 - √7)(2 + √7) = 4 - 7 = -3
Result: (10 + 5√7)/(-3) =
- (10 + 5√7)/3
Or written as:
(-10 - 5√7)/3
Either is acceptable, but let’s write with negative sign distributed:
✔ Final:
(-10 - 5√7)/3
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Now compiling all answers in order:
Final Answer:
1) 2√7
2) 6√2
3) 4√2
4) 5√7
5) 6√3
6) 4√5
7) √3
8) -21√11
9) 10√5
10) -22√21
11) √15
12) 6√6 + 27
13) 2√35
14) 2
15) 42√6
16) ∛30 / 3
17) (3 - √5 + 3√2 - √10)/4
18) (-10 - 5√7)/3
Parent Tip: Review the logic above to help your child master the concept of simplest radical form worksheet.
