If your fraction contains a sum of two terms in the denominator, at least one of which is irrational, then you cannot multiply the fraction by it in the numerator and denominator. 42+2{\displaystyle {\frac {4}{2+{\sqrt {2}}}}} To see why this is the case, write an arbitrary fraction 1a+b,{\displaystyle {\frac {1}{a+b}},} where a{\displaystyle a} and b{\displaystyle b} are irrational. Then the expression (a+b)(a+b)=a2+2ab+b2{\displaystyle (a+b)(a+b)=a^{2}+2ab+b^{2}} contains a cross-term 2ab.{\displaystyle 2ab.} If at least one of a{\displaystyle a} and b{\displaystyle b} is irrational, then the cross-term will contain a radical. Let's see how this works with our example. 42+2⋅2+22+2=4(2+2)4+42+2{\displaystyle {\frac {4}{2+{\sqrt {2}}}}\cdot {\frac {2+{\sqrt {2}}}{2+{\sqrt {2}}}}={\frac {4(2+{\sqrt {2}})}{4+4{\sqrt {2}}+2}}}  As you can see, there's no way we can get rid of the 42{\displaystyle 4{\sqrt {2}}} in the denominator after doing this. The conjugate of an expression is the same expression with the sign reversed. For example, the conjugate of 2+2{\displaystyle 2+{\sqrt {2}}} is 2−2.{\displaystyle 2-{\sqrt {2}}.}  42+2⋅2−22−2{\displaystyle {\frac {4}{2+{\sqrt {2}}}}\cdot {\frac {2-{\sqrt {2}}}{2-{\sqrt {2}}}}} Why does the conjugate work? Going back to our arbitrary fraction 1a+b,{\displaystyle {\frac {1}{a+b}},} multiplying by the conjugate in the numerator and denominator results in the denominator being (a+b)(a−b)=a2−b2.{\displaystyle (a+b)(a-b)=a^{2}-b^{2}.} The key here is that there are no cross-terms. Since both of these terms are being squared, any square roots will be eliminated. 42+2⋅2−22−2=4(2−2)4−2=4−22{\displaystyle {\frac {4}{2+{\sqrt {2}}}}\cdot {\frac {2-{\sqrt {2}}}{2-{\sqrt {2}}}}={\frac {4(2-{\sqrt {2}})}{4-2}}=4-2{\sqrt {2}}}

Summary: Examine the fraction. Multiply the fraction by the conjugate of the denominator. Simplify as needed.


Pull six inches of wired ribbon from your spool and pinch it tightly between the thumb and forefinger of your left hand. Pull some more ribbon from your spool with your right hand and form a loop.  Meet your hands together and scrunch the bottom of your loop with the rest of the pinched ribbon between the fingers of your left hand.  Twist the pinched ribbon in your left hand to secure the loop.  The size of your loops will depend on the size of your tree.  For large trees, your loops should be ten to twelve inches long which means you'll need to pull at least twenty inches of ribbon to form each loop.  Use wired ribbon.  Wired ribbon will hold the shape you put it in while other ribbon will go limp. Most ribbon is one sided.  Twist your ribbon so that the decorated side will be on the outside of your next loop. Pull more ribbon from your spool with your right hand.  Form a loop that is in the opposite direction from your first loop.  Add the bottom of your second loop to the pinched ribbon in our left hand and twist it clockwise to hold it in place.  Your first two loops should form the shape of an eight.  Make sure your second loop is the same size as the first.  You want all your loops to be uniform. At this point you can hold your loops against your tree to make sure your bow will be the right size for it. Continue making alternating loops until you feel like your bow is complete.  Ten loops should be adequate for an elaborate bow, but you can make less or more depending on your personal preference.  Remember to pinch and twist the bottom of each loop tightly.  You should have the bottoms of all the loops in your bow scrunched up between the thumb and forefinger of your left hand. The decorated side of the ribbon should be on the outside of each one of your loops. After you've made all your loops, make one final loop perpendicular to the rest.  Pinch the bottom of it in your left hand and twist it just like you've done with all the other loops. Mold a piece of floral stem wire into a “u” shape.  Insert the loose ends into your central loop and bend them around the bottom part of your bow.  Pull the loose ends through the “u” part of the wire.  Pull the loose ends in opposite directions to make a tight knot.  Twist the ends of the wire several times to make sure they are securely fastened.  Take another piece of floral wire and use the same method to fasten your bow  to the top of your Christmas tree.  Floral stem wire comes in a green, silver or white.  Make sure to pick the best color to match your bow. The lower the gauge of floral wire, the thicker and less moldable it is.  Twenty-six gauge floral stem wire is recommended for fastening bows.
Summary: Make a loop. Make a figure eight. Fill out your bow. Make your central loop. Fasten your loops together.