Math worksheet for practicing addition and subtraction of radical expressions.

Let’s solve each problem step by step. We’ll simplify radicals first, then combine like terms (only radicals with the same index and radicand can be added or subtracted).

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Problem 1: -√12 + ∛3

- √12 = √(4×3) = 2√3 → so -√12 = -2√3
- ∛3 cannot be simplified further.
- These are different types of radicals (square root vs cube root), so they cannot be combined.

→ Final: -2√3 + ∛3

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Problem 2: ∛3 - √12

Same as above:
- √12 = 2√3
- So expression becomes: ∛3 - 2√3
- Again, different radical types — cannot combine.

→ Final: ∛3 - 2√3

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Problem 3: 2√11 + 4√11

These are like terms (same radical: √11)

→ 2 + 4 = 6 → 6√11

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Problem 4: -∛27 - ∛8

Simplify each cube root:

- ∛27 = 3 → so -∛27 = -3
- ∛8 = 2 → so -∛8 = -2

→ -3 - 2 = -5

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Problem 5: √5 + 32 - √200

Wait — let’s check notation:

It says:
√5 + ∜32 - ⁺√200 → probably meant to be fifth root? The symbol looks like “⁵√” but written oddly.

Looking at context: Problem 5 has “⁵√32” and “⁵√200” — yes, it's fifth roots.

So:

- √5 stays as is.
- ∜32 → wait, no — if it’s “⁵√32”, that’s fifth root of 32.

Actually, looking again: In problem 5, it’s written as:

> ⑤ √5 + ⁵√32 - ⁵√200

Yes — those are fifth roots.

Now simplify:

- ⁵√32: 32 = 2⁵ → so ⁵√32 = 2
- ⁵√200: 200 = 32 × 6.25? Not a perfect fifth power. Let’s factor: 200 = 2³ × 5² → no fifth powers inside → cannot simplify.

Wait — actually, 200 is not a perfect fifth power. But maybe we can write it as:

⁵√200 = ⁵√(32 × 6.25)? No, better to leave as is unless it simplifies.

But 32 is 2⁵ → so ⁵√32 = 2.

200 = 2³ × 5² → no fifth power factors → so ⁵√200 stays.

So expression becomes:

√5 + 2 - ⁵√200

No like terms → cannot combine further.

→ Final: √5 + 2 - ⁵√200

Wait — but maybe I misread the original. Let me double-check the image description.

In the user’s text:
“⑤ √5 + ⁵√32 - ⁵√200”

Yes — so:

⁵√32 = 2
⁵√200 → doesn’t simplify nicely.

But perhaps 200 was meant to be something else? Or maybe we’re supposed to leave it.

Alternatively, maybe it’s a typo and should be √200? But no — the superscript 5 is there.

We’ll go with what’s given.

→ Answer: √5 + 2 - ⁵√200

But let’s hold on — maybe in some contexts, they expect you to recognize that 200 isn't simplifiable, so just leave it.

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Problem 6: -∛3 - √8 - ∛3

Group like terms:

Cube roots: -∛3 - 3 = -23
Square root: -√8 = -√(4×2) = -2√2

So total: -2∛3 - 2√2

Cannot combine further.

→ Final: -2∛3 - 2√2

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Problem 7: -√27 - ∛45 - √20 + √45

First, simplify each radical:

- √27 = √(9×3) = 3√3 → so -√27 = -3√3
- ∛45 → 45 = 9×5 → no perfect cubes → stays as ∛45
- √20 = √(4×5) = 2√5 → so -√20 = -2√5
- √45 = √(9×5) = 3√5

Now rewrite expression:

-3√3 - ∛45 - 2√5 + 3√5

Combine like terms:

For √5: -2√5 + 3√5 = +1√5 = √5

Others don’t match:

→ Final: -3√3 - ∛45 + √5

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Problem 8: ∛5 + 5 + √5

Like terms: √5 + 5 = 2√5
∛5 is different → stays separate.

→ Final: ∛5 + 2√5

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Problem 9: ∛24 - √54 + √6

Simplify each:

- ∛24: 24 = 8×3 = 2³×3 → so ∛24 = ∛(8×3) = 2∛3
- √54 = √(9×6) = 3√6 → so -√54 = -3√6
- √6 stays

Expression becomes:

2∛3 - 3√6 + √6

Combine √6 terms: -3√6 + √6 = -2√6

→ Final: 2∛3 - 2√6

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Problem 10: ∛27 + ∜20 - √300

Simplify:

- ∛27 = 3
- ∜20: 20 = 4×5 → no fourth powers → stays as ∜20
- √300 = √(100×3) = 10√3 → so -√300 = -10√3

Expression: 3 + ∜20 - 10√3

No like terms → cannot combine.

→ Final: 3 + ∜20 - 10√3

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Now, let’s compile all final answers clearly.

Final Answer:

1. -2√3 + ∛3
2. ∛3 - 2√3
3. 6√11
4. -5
5. √5 + 2 - ⁵√200
6. -2∛3 - 2√2
7. -3√3 - ∛45 + √5
8. ∛5 + 2√5
9. 2∛3 - 2√6
10. 3 + ∜20 - 10√3