Question: Samuel has tried to work out $6\sqrt{30} \div 2$ but has made a mistake.
a) Work out $6\sqrt{30} \div 2$
Give your answer in its simplest form.
b) Write a sentence to explain the mistake that Samuel has made.
$6\sqrt{30} \div 2 = \frac{6}{2} \times \sqrt{\frac{30}{2}} = 3\sqrt{15}$
1. **State the problem:** Simplify the expression $6\sqrt{30} \div 2$ correctly.
2. **Samuel's mistake:** He wrote $6\sqrt{30} \div 2$ as $\frac{6}{2} \times \sqrt{\frac{30}{2}}$, which is incorrect because division does not distribute inside the square root like that.
3. **Correct approach:** The division by 2 applies to the entire product $6\sqrt{30}$, so we write:
$$6\sqrt{30} \div 2 = \frac{6\sqrt{30}}{2}$$
4. **Simplify the fraction:**
$$\frac{6\sqrt{30}}{2} = \frac{\cancel{6}}{\cancel{2}} \times \sqrt{30} = 3\sqrt{30}$$
5. **Final answer:**
$$3\sqrt{30}$$
6. **Explanation of the mistake:** Samuel incorrectly divided the number inside the square root by 2, which is not allowed. Division outside the square root cannot be distributed inside the root.
**Summary:**
- Correct simplification: $6\sqrt{30} \div 2 = 3\sqrt{30}$
- Samuel's error: $6\sqrt{30} \div 2 \neq 3\sqrt{15}$