## iwanttogotostanford one year ago Simplify square root of negative 48.

1. anonymous

Can you think of any perfect squares that are factors of 48?

2. iwanttogotostanford

no not from the top of my head

3. iwanttogotostanford

its $\sqrt{-48}$

4. anonymous

I understand, but there's a method to my madness. You're going to have to simplify this radical, so you'll need to do this. The first few prefect squares are 1, 4, 9, 16, 25, 36, 49, etc. What is the largest of these that is a factor of 48?

5. iwanttogotostanford

4? because 4x12 is 48

6. anonymous

That's right, but is there a larger one?

7. iwanttogotostanford

I'm really rusty on my times table facts

8. anonymous

Use a calculator if it helps

9. iwanttogotostanford

ok

10. iwanttogotostanford

24

11. anonymous

Sorry, 24 is not a perfect square. I listed them above. What's the largest one that is a factor of 48?

12. iwanttogotostanford

I'm confused a bit, wouldn't it be 4? because 9x9 is 81 and thats too big

13. anonymous

9 is a perfect square because 3 x 3 = 9 16 is a perfect square because 4 x 4 = 16 25 is a perfect square because 5 x 5 = 25 etc. Which of those listed numbers is the largest one that is a factor of 48? You don't need to square them.

14. iwanttogotostanford

36

15. anonymous

Excellent. 16 x 3 = 48.

16. iwanttogotostanford

now what?

17. anonymous

Now, we're going to use the rules of working with radicals to simplify. Your question is$\sqrt{-48}$Having identified the largest perfect square that is a factor of 48 we can rewrite as follows$\sqrt{-48}=\sqrt{\left( 16 \right)\left( -1 \right)\left( 3 \right)}$Understand what we did here?

18. iwanttogotostanford

yes

19. anonymous

Good. Now the rules of radicals say that we can write this as follows$\sqrt{-48} = \sqrt{\left( 16 \right)\left( -1 \right)\left( 3 \right)} = \sqrt{16}\sqrt{-1}\sqrt{3}$You OK with that?

20. iwanttogotostanford

ok

21. anonymous

Good. You know what the square root of 16 is? And the square root of -1?

22. iwanttogotostanford

yes its 4 but i don't know the square root of -1

23. anonymous

You haven't studied imaginary numbers?

24. iwanttogotostanford

no, I'm learning them right now thats why i need help

25. anonymous

Well imaginary numbers are based on the square root of -1. It is an imaginary number that is given the symbol i. In other words$\sqrt{-1} = i$

26. anonymous

So, you have $$\sqrt{16} \sqrt{-1} \sqrt{3}$$. And you know the square root of 16 and the square root of -1. Just substitute them in.

27. iwanttogotostanford

these are my answer choices negative 4 square root of 3 4 square root of negative 3 4 i square root of 3 4 square root of 3 i

28. anonymous

$\sqrt{16}\sqrt{-1}\sqrt{3} = ?$

29. iwanttogotostanford

would it be b or d?

30. anonymous

$$\sqrt{16}=4$$ and $$\sqrt{-1} = i$$ and $$\sqrt{3}$$ can't be simplified any further. What does that give you?

31. iwanttogotostanford

D!?

32. anonymous

Well, the individual parts of D are correct but they're in the wrong order. Should have the rational number first, then the imaginary number i, then the radical. What other choice meets this description?

33. iwanttogotostanford

C then?

34. anonymous

That's correct. It's the convention that we write the answer $$4i\sqrt{3}$$ rather than $$4\sqrt{3}i$$ or $$\sqrt{3}i4$$ or any other combination.

35. iwanttogotostanford

ok, thank you i was confused but now i understand better

36. anonymous

You're welcome