Collection of simplified radical expressions and fractions with square roots.
Educational worksheet: Surds-Rationalising the denominator (worksheet with detailed solutions). Download and print for classroom or home learning activities.
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Step-by-step solution for: Surds-Rationalising the denominator (worksheet with detailed solutions)
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Show Answer Key & Explanations
Step-by-step solution for: Surds-Rationalising the denominator (worksheet with detailed solutions)
Let’s go through each expression one by one and simplify or evaluate them as needed. Since no specific instruction was given (like “simplify,” “compare,” or “rationalize”), I’ll assume the task is to simplify each radical expression if possible, or leave it in simplest form if already simplified.
We’ll check each:
---
1. $\frac{\sqrt{6}}{2}$
→ √6 cannot be simplified further (6 = 2×3, no perfect squares). Denominator is rational.
✔ Already simplified.
---
2. $\frac{\sqrt{6}}{3}$
→ Same as above — √6 is simplified, denominator is rational.
✔ Already simplified.
---
3. $\sqrt{5}$
→ 5 is prime → can’t simplify.
✔ Already simplified.
---
4. $\frac{3\sqrt{2}}{2}$
→ √2 is simplified, coefficient 3 and denominator 2 are fine.
✔ Already simplified.
---
5. $\frac{\sqrt{6}}{3}$
→ Same as #2.
✔ Already simplified.
---
6. $\frac{2\sqrt{3}}{3}$
→ √3 is simplified, numerator has coefficient 2, denominator 3 — nothing to reduce.
✔ Already simplified.
---
7. $-\frac{4\sqrt{5}}{5}$
→ √5 is simplified, fraction is reduced (4 and 5 share no common factors), negative sign is fine.
✔ Already simplified.
---
8. $-\sqrt{3}$
→ √3 is simplified, negative sign doesn’t change that.
✔ Already simplified.
---
9. $\frac{3\sqrt{2}}{8}$
→ √2 is simplified, 3 and 8 have no common factors.
✔ Already simplified.
---
10. $\frac{2\sqrt{5}}{5}$
→ √5 is simplified, 2 and 5 coprime.
✔ Already simplified.
---
Wait — maybe the task was to match equivalent expressions? Let’s look again at the list:
Left column:
- $\frac{\sqrt{6}}{2}$
- $\sqrt{5}$
- $\frac{\sqrt{6}}{3}$
- $-\frac{4\sqrt{5}}{5}$
- $\frac{3\sqrt{2}}{8}$
Right column:
- $\frac{\sqrt{6}}{3}$ ← same as left #3
- $\frac{3\sqrt{2}}{2}$
- $\frac{2\sqrt{3}}{3}$
- $-\sqrt{3}$
- $\frac{2\sqrt{5}}{5}$
Hmm — perhaps the original problem was to pair matching expressions? But only one pair matches exactly: $\frac{\sqrt{6}}{3}$ appears on both sides.
Alternatively, maybe the task was to rationalize denominators? But none of these have radicals in the denominator — they’re all already rationalized.
Another possibility: evaluate numerically? But that seems unlikely without context.
Wait — let me double-check if any can be simplified further.
For example: Is $\frac{2\sqrt{5}}{5}$ equal to something else? No.
Is $-\frac{4\sqrt{5}}{5}$ related to $\frac{2\sqrt{5}}{5}$? Only by factor of -2, but not equal.
Perhaps the real task was to identify which expressions are in simplest radical form — and the answer is: all of them are.
But since the user didn’t specify the exact question, and just said “solve the problem”, and provided a list of 10 expressions, I think the most reasonable interpretation is:
> Simplify each expression if possible.
And after checking each one carefully — none can be simplified further. They are all already in simplest form.
So the final answer is simply listing them as-is, confirming they’re simplified.
But wait — maybe the problem was to group or match? Looking again:
Left side has 5 items, right side has 5 items. Maybe it's a matching exercise?
Let’s compare values numerically to see if any are equal:
Compute approximate decimal values:
Left:
1. $\frac{\sqrt{6}}{2} ≈ \frac{2.449}{2} ≈ 1.2247$
2. $\sqrt{5} ≈ 2.236$
3. $\frac{\sqrt{6}}{3} ≈ \frac{2.449}{3} ≈ 0.8165$
4. $-\frac{4\sqrt{5}}{5} ≈ -\frac{4*2.236}{5} ≈ -\frac{8.944}{5} ≈ -1.7888$
5. $\frac{3\sqrt{2}}{8} ≈ \frac{3*1.414}{8} ≈ \frac{4.242}{8} ≈ 0.530$
Right:
1. $\frac{\sqrt{6}}{3} ≈ 0.8165$ ← matches left #3
2. $\frac{3\sqrt{2}}{2} ≈ \frac{4.242}{2} ≈ 2.121$
3. $\frac{2\sqrt{3}}{3} ≈ \frac{2*1.732}{3} ≈ \frac{3.464}{3} ≈ 1.1547$
4. $-\sqrt{3} ≈ -1.732$
5. $\frac{2\sqrt{5}}{5} ≈ \frac{4.472}{5} ≈ 0.8944$
Only exact match is $\frac{\sqrt{6}}{3}$ appearing on both sides.
No other pairs match numerically.
Perhaps the task was to write each in simplest form, and since all are already simple, we just confirm that.
Given the ambiguity, but following the instruction to “solve the problem accurately”, and since no operation was specified, I will conclude that each expression is already in its simplest form, and no further simplification is needed.
Final Answer: All expressions are already in simplest radical form.
We’ll check each:
---
1. $\frac{\sqrt{6}}{2}$
→ √6 cannot be simplified further (6 = 2×3, no perfect squares). Denominator is rational.
✔ Already simplified.
---
2. $\frac{\sqrt{6}}{3}$
→ Same as above — √6 is simplified, denominator is rational.
✔ Already simplified.
---
3. $\sqrt{5}$
→ 5 is prime → can’t simplify.
✔ Already simplified.
---
4. $\frac{3\sqrt{2}}{2}$
→ √2 is simplified, coefficient 3 and denominator 2 are fine.
✔ Already simplified.
---
5. $\frac{\sqrt{6}}{3}$
→ Same as #2.
✔ Already simplified.
---
6. $\frac{2\sqrt{3}}{3}$
→ √3 is simplified, numerator has coefficient 2, denominator 3 — nothing to reduce.
✔ Already simplified.
---
7. $-\frac{4\sqrt{5}}{5}$
→ √5 is simplified, fraction is reduced (4 and 5 share no common factors), negative sign is fine.
✔ Already simplified.
---
8. $-\sqrt{3}$
→ √3 is simplified, negative sign doesn’t change that.
✔ Already simplified.
---
9. $\frac{3\sqrt{2}}{8}$
→ √2 is simplified, 3 and 8 have no common factors.
✔ Already simplified.
---
10. $\frac{2\sqrt{5}}{5}$
→ √5 is simplified, 2 and 5 coprime.
✔ Already simplified.
---
Wait — maybe the task was to match equivalent expressions? Let’s look again at the list:
Left column:
- $\frac{\sqrt{6}}{2}$
- $\sqrt{5}$
- $\frac{\sqrt{6}}{3}$
- $-\frac{4\sqrt{5}}{5}$
- $\frac{3\sqrt{2}}{8}$
Right column:
- $\frac{\sqrt{6}}{3}$ ← same as left #3
- $\frac{3\sqrt{2}}{2}$
- $\frac{2\sqrt{3}}{3}$
- $-\sqrt{3}$
- $\frac{2\sqrt{5}}{5}$
Hmm — perhaps the original problem was to pair matching expressions? But only one pair matches exactly: $\frac{\sqrt{6}}{3}$ appears on both sides.
Alternatively, maybe the task was to rationalize denominators? But none of these have radicals in the denominator — they’re all already rationalized.
Another possibility: evaluate numerically? But that seems unlikely without context.
Wait — let me double-check if any can be simplified further.
For example: Is $\frac{2\sqrt{5}}{5}$ equal to something else? No.
Is $-\frac{4\sqrt{5}}{5}$ related to $\frac{2\sqrt{5}}{5}$? Only by factor of -2, but not equal.
Perhaps the real task was to identify which expressions are in simplest radical form — and the answer is: all of them are.
But since the user didn’t specify the exact question, and just said “solve the problem”, and provided a list of 10 expressions, I think the most reasonable interpretation is:
> Simplify each expression if possible.
And after checking each one carefully — none can be simplified further. They are all already in simplest form.
So the final answer is simply listing them as-is, confirming they’re simplified.
But wait — maybe the problem was to group or match? Looking again:
Left side has 5 items, right side has 5 items. Maybe it's a matching exercise?
Let’s compare values numerically to see if any are equal:
Compute approximate decimal values:
Left:
1. $\frac{\sqrt{6}}{2} ≈ \frac{2.449}{2} ≈ 1.2247$
2. $\sqrt{5} ≈ 2.236$
3. $\frac{\sqrt{6}}{3} ≈ \frac{2.449}{3} ≈ 0.8165$
4. $-\frac{4\sqrt{5}}{5} ≈ -\frac{4*2.236}{5} ≈ -\frac{8.944}{5} ≈ -1.7888$
5. $\frac{3\sqrt{2}}{8} ≈ \frac{3*1.414}{8} ≈ \frac{4.242}{8} ≈ 0.530$
Right:
1. $\frac{\sqrt{6}}{3} ≈ 0.8165$ ← matches left #3
2. $\frac{3\sqrt{2}}{2} ≈ \frac{4.242}{2} ≈ 2.121$
3. $\frac{2\sqrt{3}}{3} ≈ \frac{2*1.732}{3} ≈ \frac{3.464}{3} ≈ 1.1547$
4. $-\sqrt{3} ≈ -1.732$
5. $\frac{2\sqrt{5}}{5} ≈ \frac{4.472}{5} ≈ 0.8944$
Only exact match is $\frac{\sqrt{6}}{3}$ appearing on both sides.
No other pairs match numerically.
Perhaps the task was to write each in simplest form, and since all are already simple, we just confirm that.
Given the ambiguity, but following the instruction to “solve the problem accurately”, and since no operation was specified, I will conclude that each expression is already in its simplest form, and no further simplification is needed.
Final Answer: All expressions are already in simplest radical form.
Parent Tip: Review the logic above to help your child master the concept of rationalizing the denominator worksheets.
