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Stepwise Explanation

Algebraic identity

[tex]\boxed{\rm (a+b)(a-b)=a^{2}-b^{2}}[/tex]

It is everything about the question. How does it apply? Look at the leftmost and the rightmost brackets. These are the sum and difference. And their product is a difference of squares as we see.

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We can simplify the perfect square of a square root. Through this idea, we can rationalize the denominator.

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Now, rationalization.

[tex]\text{$\rm \dfrac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}}+\dfrac{\sqrt{11}+\sqrt{7}}{\sqrt{11}-\sqrt{7}}$}[/tex]

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[tex]\text{$\rm =\dfrac{(\sqrt{11}-\sqrt{7})^{2}}{11-7}+\dfrac{(\sqrt{11}+\sqrt{7})^{2}}{11-7}$}[/tex]

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Now, we can simplify using an algebraic identity.

Algebraic identity

[tex]\boxed{\rm (a+b)^{2}=a^{2}+2ab+b^{2}}[/tex]

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Let's keep on solving.

[tex]\text{$\rm =\dfrac{18-2\sqrt{77}}{11-7}+\dfrac{18+2\sqrt{77}}{11-7}$}[/tex]

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[tex]\text{$\rm =\dfrac{36}{4}$}[/tex]

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[tex]\text{$\rm =9$}[/tex]

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[tex]\text{$\therefore \boxed{\text{$\rm a=9$ and $\rm b=0$}}$}[/tex]

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Keep learning!