### Video Transcript

Simplify nine divided by root 13 plus root seven plus nine divided by root 13 minus root seven.

Our first step is to try and rationalize the denominator by finding a common denominator. We do this by multiplying root 13 plus root seven by root 13 minus root seven. As we have multiplied the denominator of the first fraction by root 13 minus root seven, we need to multiply the numerator by the same thing. In the same way, we need to multiply the numerator of the second fraction by root 13 plus root seven.

Expanding the first parenthesis gives us nine root 13 minus nine root seven and expanding the second parenthesis or bracket gives us nine root 13 plus nine root seven. The denominator root 13 plus root seven multiplied by root 13 minus root seven is the difference of two squares. This means that when we expanded using the FOIL method, the outside and the inside terms will cancel.

This means we only need to multiply the first terms root 13 by root 13 and the last terms root seven by negative root seven. Root 13 multiplied by root 13 is equal to 13 and positive root seven multiplied by negative root seven is equal to negative seven. 13 minus seven is equal to six. Therefore, our denominator is six. This leaves us with nine root 13 minus nine root seven plus nine root 13 plus nine root seven.

Simplifying the numerator gives us 18 root 13 as nine root 13 plus nine root 13 is equal to 18 root 13. And negative nine root seven plus nine root seven is equal to zero. 18 divided by six is equal to three. Therefore, our final answer is three root 13.

Nine divided by root 13 plus root seven plus nine divided by root 13 minus root seven is equal to three root 13.