How does √3 + 3√3 = 4√3

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nav1

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Hello

I don't understand how or why √3 + 3√3 = 4√3

How does it equal 4√3?

My understanding was √3 + √3 = 2√3, but when I add the 3 (before the radical) how do we get to 4√3?

Can you please provide steps in your answer, so I understand how to solve in the future

Thank you very much!
 
note [MATH]\sqrt{3} = 1 \cdot \sqrt{3}[/MATH]
[MATH]1 \cdot \sqrt{3} + 3 \cdot \sqrt{3} = (1 + 3) \cdot \sqrt{3} = 4 \cdot \sqrt{3}[/MATH]
 
nav1 said:
Hello

I don't understand how or why √3 + 3√3 = 4√3

How does it equal 4√3?

My understanding was √3 + √3 = 2√3, but when I add the 3 (before the radical) how do we get to 4√3?

Can you please provide steps in your answer, so I understand how to solve in the future

Thank you very much!
Click to expand...
If you add the 3 to √3 → will make 3 + √3

You should know that

3√3 = 3 * √3​

Does that clarify your doubt?
 
lev888 said:
Replace x with `sqrt(3)`:
`2sqrt(3) + 3sqrt(3) = 5sqrt(3)`
Click to expand...
Ah I see.

So the √3 doesn't change.

But what's confuses me is, √3 + √3 = 2√3
Then we add the 3 before the radical to get 4√3 how is that possible?
 
nav1 said:
Ah I see.

So the √3 doesn't change.

But what's confuses me is, √3 + √3 = 2√3
Then we add the 3 before the radical to get 4√3 how is that possible?
Click to expand...
My example is different from your problem: I used 2x instead 1x. But the same idea applies - the distributive property of multiplication: a(b+c) = ab + ac.
 
nav1 said:
Ah I see.

So the √3 doesn't change.

But what's confuses me is, √3 + √3 = 2√3
Then we add the 3 before the radical to get 4√3 how is that possible?
Click to expand...
It seems you are not interpreting that expression correctly. There is no √3 + √3 there. We have √3 + 3√3, which is equivalent to x+3x with x being √3.
x+3x = 4x = 4√3.
 
nav1 said:
Ah I see.

So the √3 doesn't change.

But what's confuses me is, √3 + √3 = 2√3
Then we add the 3 before the radical to get 4√3 how is that possible?
Click to expand...
An apple plus an apple is 2 apples.

What is an apple plus 3 apples?
 
Dr.Peterson said:
So what is a √3 plus 3 √3's?
Click to expand...
Going by lev888's previous comment, I understand when replacing the radical with an x.

He said
"We have √3 + 3√3, which is equivalent to x+3x with x being √3.
x+3x = 4x = 4√3."

Yes that makes sense. I struggle to see the radical symbol as simple as x is.

x+3x = 4x. No problem

However, √3 + 3√3 is confusing. I keep wondering if the radical is added or divided with the 3.
 
nav1 said:
Going by lev888's previous comment, I understand when replacing the radical with an x.

He said
"We have √3 + 3√3, which is equivalent to x+3x with x being √3.
x+3x = 4x = 4√3."

Yes that makes sense. I struggle to see the radical symbol as simple as x is.

x+3x = 4x. No problem

However, √3 + 3√3 is confusing. I keep wondering if the radical is added or divided with the 3.
Click to expand...
Where do you see the second "+" sign? Or division? 3x is multiplication, not matter what x is.
 
lev888 said:
Where do you see the second "+" sign? Or division? 3x is multiplication, not matter what x is.
Click to expand...
You're right. Thank you for helping me understand this.

Upon reflection that was an embarrassing assertion I made. The multiplication is implied. I understand now.

Can you recommend a site where I can practise adding radicals?

Thank you again!
 
nav1 said:
You're right. Thank you for helping me understand this.

Upon reflection that was an embarrassing assertion I made. The multiplication is implied. I understand now.

Can you recommend a site where I can practise adding radicals?

Thank you again!
Click to expand...
Welcome!
www.purplemath.com

How to Add and Subtract with Square Roots

To simplify square-root expressions, combine the coefficients of terms that contain the same square-root part; e.g., 2√[3]+3√[3] = 5√[3].
www.purplemath.com www.purplemath.com
 
Thank you all again for your comments, I was losing my mind with this problem.
I understand now.

Regards, nav1
 
Equivalently, \(\displaystyle 3\sqrt{3}= \sqrt{3}+ \sqrt{3}+ \sqrt{3}\)
so\(\displaystyle \sqrt{3}+ 3\sqrt{3}= \sqrt{3}+ \sqrt{3}+ \sqrt{3}+ \sqrt{3}= 4\sqrt{3}\).
 
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