Let’s solve each problem step by step. We’ll simplify radicals first, then combine like terms.
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Problem 1: 5√3 – 3√3 + 4√8
Step 1: Simplify √8
√8 = √(4×2) = √4 × √2 = 2√2
So 4√8 = 4 × 2√2 = 8√2
Step 2: Combine like radicals
5√3 – 3√3 = (5–3)√3 = 2√3
We can’t combine 2√3 and 8√2 — different radicands.
Final:
2√3 + 8√2
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Problem 2: 11√28 + √7 + 2√7
Step 1: Simplify √28
√28 = √(4×7) = √4 × √7 = 2√7
So 11√28 = 11 × 2√7 = 22√7
Step 2: Combine all √7 terms
22√7 + √7 + 2√7 = (22+1+2)√7 = 25√7
Final:
25√7
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Problem 3: 24√5 – 9√5
Same radical → just subtract coefficients
24 – 9 = 15
Final:
15√5
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Problem 4: –3√11 – 2√44 – 5√10
Step 1: Simplify √44
√44 = √(4×11) = √4 × 11 = 2√11
So –2√44 = –2 × 2√11 = –4√11
Step 2: Combine like radicals
–3√11 – 4√11 = –7√11
Can’t combine with –5√10
Final:
–7√11 – 5√10
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Problem 5: √18 + 8√27 – 7√3
Step 1: Simplify √18 and √27
√18 = √(9×2) = 3√2
√27 = √(9×3) = 3√3 → so 8√27 = 8 × 3√3 = 24√3
Step 2: Rewrite expression
3√2 + 24√3 – 7√3
Step 3: Combine √3 terms
24√3 – 7√3 = 17√3
Final:
3√2 + 17√3
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Problem 6: –2√20 + 2√45 – √5
Step 1: Simplify √20 and √45
√20 = √(4×5) = 2√5 → so –2√20 = –2 × 2√5 = –4√5
√45 = √(9×5) = 3√5 → so 2√45 = 2 × 3√5 = 6√5
Step 2: Rewrite expression
–4√5 + 6√5 – √5
Step 3: Combine
(–4 + 6 – 1)√5 = 1√5 = √5
Final:
√5
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Problem 7: –9√15 + 10√15
Same radical → add coefficients
–9 + 10 = 1
Final:
√15
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Problem 8: –3√24 – √6
Step 1: Simplify √24
√24 = √(4×6) = 2√6
So –3√24 = –3 × 2√6 = –6√6
Step 2: Combine with –√6
–6√6 – √6 = –7√6
Final:
–7√6
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Final Answer:
1) 2√3 + 8√2
2) 25√7
3) 15√5
4) –7√11 – 5√10
5) 3√2 + 17√3
6) √5
7) √15
8) –7√6
Parent Tip: Review the logic above to help your child master the concept of operations with radical expressions worksheet answers.
