Hello! If you can help me with these 3 questions I would love it!

Question

1. How do you simplify sqrt5 + 6 sqrt 5? I think that it is 5 sqrt10. Am I correct?
2. 5 sqrt 7 +2 sqrt 175. I think the answer is 7 sqrt 182
3. What is the domain of this function y = sqrt 3x+3. I really need help on this one.

Can someone just explain this too me? I don't want to just get the answers. I want to learn it. Thanks again!

The Great Undertale Fanatic

Answer 1

sqrt5 + 6 sqrt 5
= √5 + 6√5
= 1√5 + 6√5

1 apple + 6 apples = 7 apples
1 coffee + 6 coffees = 7 coffees
1x + 6x = 7x
1√5 + 6√5 = 7√5

You can easily verify this with your calculator
√5 = 2.236067977..
6√5 = 13.41640786..
so √5 + 6√5 = 2.236067977.. + 13.41640786.. = 15.65247584... (my calculator rounded)
7√5 = 15.65247584...
but your 5√10 answer would be 15.811388.. ≠ 7√5

for 5√7 +2√175
we cannot add them they way they sit, just like we couldn't add 5x + 2y
BUT ....
√175 = (√25)(√7) = 5√7 , we can only do this if one or more of the factors
are perfect squares.
5√7 +2√175
= 5√7 + 5√7
= 10√7 , (same as in our previous question).
again, check with your calculator

for y = √( 3x+3)
remember we cannot take the square root of a negative, so
3x+3 ≥ 0
divide both sides by +3, (nothing happens to the ≥ sign)
x + 3 ≥ 0
subtract 3 from both sides:
x ≥ -3 <------- your domain

Answer 2

Thank you for helping me on this!

Answer 3

You are welcome