Question

1. determine if the product of $3sqrt{2}$ and $4sqrt{2}$ is rational or irrational. explain your answer.
2. $-4sqrt{15} cdot -sqrt{3}$

activity: find your way through the maze on the next page. use the space provided below to show your work.

1. $dfrac{sqrt{50}}{sqrt{2}}$

Explanation:

Step1: Simplify Q6: Combine radicals

$\frac{\sqrt{50}}{\sqrt{2}} = \sqrt{\frac{50}{2}}$

Step2: Calculate fraction inside radical

$\sqrt{\frac{50}{2}} = \sqrt{25}$

Step3: Evaluate square root

$\sqrt{25} = 5$

Step4: Solve Q9: Multiply coefficients/radicals

$3\sqrt{2} \times 4\sqrt{2} = (3 \times 4) \times (\sqrt{2} \times \sqrt{2})$

Step5: Compute product components

$12 \times (\sqrt{2^2}) = 12 \times 2$

Step6: Final product & classify

$12 \times 2 = 24$. 24 is rational (can be written as $\frac{24}{1}$).

Step7: Solve Q10: Multiply negatives/terms

$-4\sqrt{15} \times -\sqrt{3} = (4 \times 1) \times (\sqrt{15} \times \sqrt{3})$

Step8: Combine radicals & simplify

$4 \times \sqrt{15 \times 3} = 4 \times \sqrt{45} = 4 \times \sqrt{9 \times 5}$

Step9: Simplify radical fully

$4 \times 3\sqrt{5} = 12\sqrt{5}$

Answer:

1. $5$
2. The product is 24, which is rational because it can be expressed as a ratio of two integers ($\frac{24}{1}$).
3. $12\sqrt{5}$