Extreme Rationalizing!

Algebra Level 5

10+1+10+2++10+99101+102++1099\large\dfrac{\sqrt{\color{#20A900}{10}\color{#3D99F6}+\sqrt{1}}\color{#3D99F6}+\sqrt{\color{#20A900}{10}\color{#3D99F6}+\sqrt{2}}\color{#3D99F6}+\cdots\color{#3D99F6}+\sqrt{\color{#20A900}{10}\color{#3D99F6}+\sqrt{99}}}{\sqrt{\color{#20A900}{10}\color{#69047E}-\sqrt{1}}\color{#3D99F6}+\sqrt{\color{#20A900}{10}\color{#69047E}-\sqrt{2}}\color{#3D99F6}+\cdots\color{#3D99F6}+\sqrt{\color{#20A900}{10}\color{#69047E}-\sqrt{99}}}

If the expression above equals to a+b a+\sqrt{b} for positive integers aa and bb, find a+ba+b.

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