Square and Cube roots of numbers | Simplifying square and cube - Free Printable

Let’s solve each problem step by step. We’ll simplify square roots where possible, then combine like terms (only if they have the same radical part).

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Row 1:

1. 4√7 + 6√7
→ Same radical: √7
→ 4 + 6 = 10
→ Answer: 10√7

2. √5 + 6√5
→ √5 is same as 1√5
→ 1 + 6 = 7
→ Answer: 7√5

3. 5√7 - 3√7
→ 5 - 3 = 2
→ Answer: 2√7

4. 2√5 + 4√5
→ 2 + 4 = 6
→ Answer: 6√5

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Row 2:

5. √6 - 4√6
→ 1√6 - 4√6 = -3√6
→ Answer: -3√6

6. √8 - √2
→ Simplify √8: √(4×2) = 2√2
→ So: 2√2 - √2 = 1√2
→ Answer: √2

7. √54 + √24
→ √54 = √(9×6) = 3√6
→ √24 = √(4×6) = 2√6
→ 3√6 + 2√6 = 5√6
→ Answer: 5√6

8. 3√20 + 2√45
→ √20 = √(4×5) = 2√5 → 3×2√5 = 6√5
→ √45 = √(9×5) = 3√5 → 2×3√5 = 6√5
→ 6√5 + 6√5 = 12√5
→ Answer: 12√5

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Row 3:

9. 2√54 - 3√6
→ √54 = 3√6 → 2×3√6 = 6√6
→ 6√6 - 3√6 = 3√6
→ Answer: 3√6

10. -√3 + 2√27
→ √27 = √(9×3) = 3√3 → 2×3√3 = 6√3
→ -√3 + 6√3 = 5√3
→ Answer: 5√3

11. -√12 - 2√12
→ Combine: -1√12 - 2√12 = -3√12
→ But simplify √12: √(4×3) = 2√3 → -3×2√3 = -6√3
→ Answer: -6√3

12. -√5 + 2√45
→ √45 = 3√5 → 2×3√5 = 6√5
→ -√5 + 6√5 = 5√5
→ Answer: 5√5

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Row 4:

13. 2√12 + 3√27
→ √12 = 2√3 → 2×2√3 = 4√3
→ √27 = 3√3 → 3×3√3 = 9√3
→ 4√3 + 9√3 = 13√3
→ Answer: 13√3

14. 2√72 - 3√50
→ √72 = √(36×2) = 6√2 → 2×6√2 = 12√2
→ √50 = √(25×2) = 5√2 → 3×5√2 = 15√2
→ 12√2 - 15√2 = -3√2
→ Answer: -3√2

15. 6√50 + 3√5
→ √50 = 5√2 → 6×5√2 = 30√2
→ 30√2 + 3√5 → Can’t combine! Different radicals.
→ Wait — that can’t be right. Let me check again.

Wait — original says: 6√50 + 3√5

But √50 = 5√2, so 6√50 = 30√2
And 3√5 stays as is.

So we have 30√2 + 3√5 — these are NOT like terms. Cannot combine.

BUT — maybe I misread? Let me double-check the problem.

Looking back at image: It says “6√50 + 3√5”

Yes — different radicals. So answer should be left as is? But that seems odd for this worksheet — probably a typo or I made mistake?

Wait — perhaps it’s 6√50 + 3√2? No, user wrote 3√5.

Alternatively — maybe it’s 6√50 + 3√50? But no.

Hold on — let me recalculate:

Actually, in row 4, column 3: “6√50 + 3√5”

But √50 = 5√2, so 6√50 = 30√2

3√5 cannot be combined with √2.

But looking at other problems, all others combine nicely. Maybe it’s meant to be 6√50 + 3√2? Or 6√5 + 3√5?

Wait — perhaps I misread the original. Let me look again.

In the image description: “6√50 + 3√5” — yes.

But that doesn’t simplify to a single term. However, in context of worksheet, likely intended to be combinable.

Wait — another thought: maybe it’s 6√50 + 3√50? But no, written as 3√5.

Perhaps it’s a trick question? Unlikely for this level.

Wait — let me check if √50 and √5 can be related? No.

Unless... did I make a mistake earlier?

No — √50 = 5√2, not involving √5.

So unless there’s a typo, we must leave as 30√2 + 3√5.

But that breaks pattern. Let me check next one.

16. 2√12 + 4√3
→ √12 = 2√3 → 2×2√3 = 4√3
→ 4√3 + 4√3 = 8√3
→ Answer: 8√3

Back to 15: 6√50 + 3√5

Perhaps it’s 6√50 + 3√2? Then 30√2 + 3√2 = 33√2 — makes sense.

Or 6√5 + 3√5 = 9√5.

But as written, it’s 6√50 + 3√5.

Given that all other problems combine, and this is likely a standard worksheet, I suspect it might be a typo, but since I must solve as given, I’ll proceed.

But wait — let me re-express:

6√50 = 6 * √(25*2) = 6*5√2 = 30√2

3√5 remains.

So final expression: 30√2 + 3√5 — cannot simplify further.

But that seems out of place. Perhaps the problem is 6√50 + 3√50? That would be 9√50 = 9*5√2 = 45√2.

Or 6√5 + 3√5 = 9√5.

Looking at the image again — in the user's text: "6√50 + 3√5"

I think I have to go with what's written.

But let me check online or think differently.

Another idea: perhaps "3√5" is meant to be "3√50"? But no.

Maybe it's 6√50 + 3√2 — then 30√2 + 3√2 = 33√2.

That would fit.

Given the context, and that all other problems combine, I'll assume it's a common error and it's meant to be 6√50 + 3√2.

But to be accurate, I should solve as written.

However, let's look at the last row.

Row 5:

17. -2√12 - √3
→ √12 = 2√3 → -2*2√3 = -4√3
→ -4√3 - √3 = -5√3
→ Answer: -5√3

18. -√8 + 3√8
→ -1√8 + 3√8 = 2√8
→ Simplify √8 = 2√2 → 2*2√2 = 4√2
→ Answer: 4√2

19. 2√5 - √20
→ √20 = 2√5
→ 2√5 - 2√5 = 0
→ Answer: 0

20. -2√27 - √27
→ -2√27 -1√27 = -3√27
→ √27 = 3√3 → -3*3√3 = -9√3
→ Answer: -9√3

Now back to problem 15: 6√50 + 3√5

If I leave it as 30√2 + 3√5, it's correct mathematically, but unusual for this worksheet.

Perhaps the problem is 6√50 + 3√50? But it's written as 3√5.

Another possibility: maybe "3√5" is "3√5" but in context, perhaps it's a different number.

I recall that in some fonts, 2 and 5 look similar, but unlikely.

Perhaps it's 6√50 + 3√2 — let me calculate that: 30√2 + 3√2 = 33√2.

That makes sense.

Given that, and to maintain consistency with the worksheet's design, I'll assume it's 6√50 + 3√2.

But the user wrote "3√5", so I should stick to that.

To resolve this, let me see if there's a standard interpretation.

Upon second thought, in the original image description, it's "6√50 + 3√5", so I must use that.

So for accuracy:

6√50 = 6 * 5√2 = 30√2

3√5 = 3√5

No common factors, so answer is 30√2 + 3√5.

But that seems incorrect for the level. Perhaps I miscalculated √50.

√50 = √(25*2) = 5√2, yes.

3√5 is already simplified.

So technically, the answer is 30√2 + 3√5.

But let's move on and come back.

For now, I'll note it.

Perhaps the problem is 6√50 + 3√50, but it's not.

Another idea: maybe "3√5" is "3 times sqrt(5)", but in the context, perhaps it's a typo and it's 3√2.

I think for the sake of this exercise, and since all other problems combine, I'll assume it's 6√50 + 3√2 = 30√2 + 3√2 = 33√2.

But to be precise, let's keep it as is for now.

Let's list all answers first.

After careful review, I think there might be a typo in the problem, but since I have to solve as given, I'll do so.

For problem 15: 6√50 + 3√5 = 30√2 + 3√5

But let's write it as is.

Perhaps the student is expected to leave it unsimplified, but that doesn't make sense.

Another thought: maybe "3√5" is "3√5" but 5 is under the root, and it's correct.

I think I have to accept it.

So for problem 15: 30√2 + 3√5

But let's confirm with calculation.

6 * sqrt(50) = 6 * 7.071 = approximately 42.426

3 * sqrt(5) = 3 * 2.236 = 6.708

Sum ≈ 49.134

30*sqrt(2) + 3*sqrt(5) = 30*1.414 + 3*2.236 = 42.42 + 6.708 = 49.128, matches.

If it were 33√2 = 33*1.414 = 46.662, not match.

So mathematically, it's 30√2 + 3√5.

But for the worksheet, perhaps it's intentional to have one that doesn't combine, but unlikely.

Looking back at the user's input: "6√50 + 3√5" — yes.

I'll proceed with the mathematical answer.

So for problem 15: 30√2 + 3√5

But let's see the final answer format; perhaps I can write it as is.

To save time, I'll complete the list.

So summarizing all:

1. 10√7

2. 7√5

3. 2√7

4. 6√5

5. -3√6

6. √2

7. 5√6

8. 12√5

9. 3√6

10. 5√3

11. -6√3

12. 5√5

13. 13√3

14. -3√2

15. 30√2 + 3√5 [as per given]

16. 8√3

17. -5√3

18. 4√2

19. 0

20. -9√3

For problem 15, upon double-checking the original image description, it is indeed "6√50 + 3√5", so I'll keep it.

But let's make sure about problem 14: 2√72 - 3√50

√72 = 6√2, so 2*6√2 = 12√2

√50 = 5√2, so 3*5√2 = 15√2

12√2 - 15√2 = -3√2, correct.

Problem 15: 6√50 + 3√5 = 30√2 + 3√5

Perhaps the answer is to leave it, or maybe factor, but no.

I think it's fine.

So final answers for each cell:

Row 1: 10√7, 7√5, 2√7, 6√5

Row 2: -3√6, √2, 5√6, 12√5

Row 3: 3√6, 5√3, -6√3, 5√5

Row 4: 13√3, -3√2, 30√2 + 3√5, 8√3

Row 5: -5√3, 4√2, 0, -9√3

For problem 15, to write it neatly, perhaps 3(10√2 + √5), but not necessary.

I think for the answer, we can list them as is.

So for the Final Answer, I'll provide the grid of answers.

Since the user asked to solve the problem, and it's a grid, I'll list the answers in order.

Final Answer:

10√7, 7√5, 2√7, 6√5,
-3√6, √2, 5√6, 12√5,
3√6, 5√3, -6√3, 5√5,
13√3, -3√2, 30√2 + 3√5, 8√3,
-5√3, 4√2, 0, -9√3

But for problem 15, let's confirm once more.

Perhaps "3√5" is "3√5" but in the context of the worksheet, it might be a mistake, but I have to go with what's given.

Another idea: maybe it's 6√50 + 3√5, and they want us to simplify each, which we did, and leave as sum.

So I think it's acceptable.

So I'll box the answers.

For the response, since no options are true, only Final Answer.

So:

Final Answer:
10√7, 7√5, 2√7, 6√5, -3√6, √2, 5√6, 12√5, 3√6, 5√3, -6√3, 5√5, 13√3, -3√2, 30√2 + 3√5, 8√3, -5√3, 4√2, 0, -9√3

To make it clear, perhaps list by rows.

Since it's a grid, but for text, comma-separated is fine.

I think that's it.