Answer:
![a+b+c=9+8+6=23](https://tex.z-dn.net/?f=a%2Bb%2Bc%3D9%2B8%2B6%3D23)
Step-by-step explanation:
Let’s first simplify our expression. We have:
![\displaystyle {\sqrt2+\frac{1}{\sqrt2}+\sqrt3+\frac{1}{\sqrt{3}}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%7B%5Csqrt2%2B%5Cfrac%7B1%7D%7B%5Csqrt2%7D%2B%5Csqrt3%2B%5Cfrac%7B1%7D%7B%5Csqrt%7B3%7D%7D)
For the second term, multiply both the numerator and denominator by √2. This yields:
![\displaystyle \frac{1}{\sqrt{2}}\cdot\frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{2}}{2}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B1%7D%7B%5Csqrt%7B2%7D%7D%5Ccdot%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B%5Csqrt%7B2%7D%7D%3D%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D)
Similarly, for the fourth term, multiply both the numerator and denominator by √3. This yields:
![\displaystyle\frac{1}{\sqrt{3}}\cdot\frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{3}}{3}](https://tex.z-dn.net/?f=%5Cdisplaystyle%5Cfrac%7B1%7D%7B%5Csqrt%7B3%7D%7D%5Ccdot%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B%5Csqrt%7B3%7D%7D%3D%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B3%7D)
Hence, our expression is now:
![\displaystyle =\sqrt{2}+\frac{\sqrt{2}}{2}+\sqrt{3}+\frac{\sqrt{3}}{3}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%3D%5Csqrt%7B2%7D%2B%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%2B%5Csqrt%7B3%7D%2B%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B3%7D)
Let’s combine them. First, we will need common denominators.
Our denominators are 2 and 3. So, our common denominator will be its LCM.
The LCM of 2 and 3 is 6.
Hence, let’s make each term’s denominator 6.
For the first term, we can multiply both layers by 6. Hence:
![\displaystyle \sqrt{2}=\frac{6\sqrt{2}}{6}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Csqrt%7B2%7D%3D%5Cfrac%7B6%5Csqrt%7B2%7D%7D%7B6%7D)
For the second term, we can multiply both layers by 3. Hence:
![\displaystyle \frac{\sqrt{2}}{2}=\frac{3\sqrt{2}}{6}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B%5Csqrt%7B2%7D%7D%7B2%7D%3D%5Cfrac%7B3%5Csqrt%7B2%7D%7D%7B6%7D)
For the third term, we can multiply both layers by 6. Hence:
![\displaystyle \sqrt{3}=\frac{6\sqrt{3}}{6}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Csqrt%7B3%7D%3D%5Cfrac%7B6%5Csqrt%7B3%7D%7D%7B6%7D)
And for the last term, we can multiply both layers by 2. Hence:
![\displaystyle \frac{\sqrt{3}}{3}=\frac{2\sqrt{3}}{6}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7B%5Csqrt%7B3%7D%7D%7B3%7D%3D%5Cfrac%7B2%5Csqrt%7B3%7D%7D%7B6%7D)
So, our expression is:
![\displaystyle =\frac{6\sqrt{2}}{6}+\frac{3\sqrt{2}}{6}+\frac{6\sqrt{3}}{6}+\frac{2\sqrt{3}}{6}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%3D%5Cfrac%7B6%5Csqrt%7B2%7D%7D%7B6%7D%2B%5Cfrac%7B3%5Csqrt%7B2%7D%7D%7B6%7D%2B%5Cfrac%7B6%5Csqrt%7B3%7D%7D%7B6%7D%2B%5Cfrac%7B2%5Csqrt%7B3%7D%7D%7B6%7D)
Add:
![\displaystyle =\frac{6\sqrt{2}+3\sqrt{2}+6\sqrt{3}+2\sqrt{3}}{6}](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%3D%5Cfrac%7B6%5Csqrt%7B2%7D%2B3%5Csqrt%7B2%7D%2B6%5Csqrt%7B3%7D%2B2%5Csqrt%7B3%7D%7D%7B6%7D)
Combine like terms:
![\displaystyle=\frac{9\sqrt{2}+8\sqrt{3}}{6}](https://tex.z-dn.net/?f=%5Cdisplaystyle%3D%5Cfrac%7B9%5Csqrt%7B2%7D%2B8%5Csqrt%7B3%7D%7D%7B6%7D)
This cannot be simplified. So, c is as small as possible.
Hence: a=9, b=8, and c=6.
Therefore:
![a+b+c=9+8+6=23](https://tex.z-dn.net/?f=a%2Bb%2Bc%3D9%2B8%2B6%3D23)