The number $(\sqrt{2}+\sqrt{3})^3$ can be written in the form $a\sqrt{2} + b\sqrt{3} + c\sqrt{6}$, where $a$, $b$, and $c$ are integers. What is $a+b+c$?

The number \((\sqrt{2}+\sqrt{3})^3\) can be written in the form \(a\sqrt{2} + b\sqrt{3} + c\sqrt{6}\), where \(a\), \(b\), and \(c\) are integers. What is \(a+b+c\)?

The number (√2+√3)³ can be written in the form a√2+b√3+c√6, where a, b, and c are integers. What is a+b+c?

i put three forms of the same question

THX EVERYBODY!

qpwoei Dec 8, 2018

#1**+2 **

The number (√2+√3)³ can be written in the form a√2+b√3+c√6, where a, b, and c are integers. What is a+b+c?

\((\sqrt2+\sqrt3)^3=(\sqrt3)^3+3(\sqrt3)^2(\sqrt2)+3(\sqrt3)(\sqrt2)^2+(\sqrt2)^3\\ \)

Now expand it out and get the answer for yourself. If you have new problems with this question then let us know.

Melody Dec 8, 2018

#4**+2 **

**Yes thanks for posting clearly**.

Here is the version I would have liked best

\(\text{The number }\;(\sqrt{2}+\sqrt{3})^3 \text{ can be written in the form }\\a\sqrt{2} + b\sqrt{3} + c\sqrt{6}, \text{ where $a$, $b$, and care integers. What is a+b+c ?}\)

I like this version best becasue it is the easiest to copy and to work on.

Melody Dec 9, 2018

#6**+2 **

I'll post a solution:

We just apply the sum of cubes formula, so, we get \(\left(\sqrt{2}\right)^3+3\left(\sqrt{2}\right)^2\sqrt{3}+3\sqrt{2}\left(\sqrt{3}\right)^2+\left(\sqrt{3}\right)^3\) .

We just simplify this and add like terms, to reach, \(11\sqrt{2}+9\sqrt{3}.\)

From here, we know that \(a=11, b=9, c=0\) , thus \(11+9+0=\boxed{20}.\)

.tertre Dec 9, 2018