...or a+^/b, are called binomial surds, and may be reduced to rational quantities on the principle that the product of the sum and difference of two quantities is equal to the difference of their squares. Thus the binomial surd v/a+v/6 Multiplied by - . v/o — -J/b — Vab+b Gives - - . a +b, n. rational...

...parts. Also, that («-|-i)(a — i)=a2 — b-. Or (a4+i4)O4— J4)=a2— b2. That is, the rectangle of the sum and difference of two quantities is equal to the difference of their squares. By this last observation we readily perceive that a4— b4, or any other binomial having a minus sign...

...multiplied into a — 6, the product will be a2— b2, (Art. 86 ;) that is, 191. The product ofthesum and difference of two quantities, is equal to the difference of their squares. This is an instance of the facility with which general truths are demonstrated in algebra. If the sum...

...square of the second. 18. Multiply a+b by a — b. The product is a2 — b2 ; whence we find, that the product of the sum, and difference of -two quantities, is equal to the difference of their squares. These results it is important to retain, as they are of frequent application. Homogeneous terms, are...

...less than the sum of their squares by twice their product. (See Art. 28, Example 2.) 31. THEOREM III. The product of the sum and difference of two quantities is equal to the difference of their squares. (See Art. 28, Example 3.) NOTE. The above theorems are very important, and should be committed to memory....

...examples afford results which should be firmly fixed in the memory. From the 1st of these we see that " The product of the sum and difference of two quantities, is equal to the difference of the squares of those quantities." From the 2nd of these we see that "The square of the sum of two quantities,...

...some other given irrational quantity, will produce a rational result : thus, V& =« V* 3 =* Again, since the product of the sum and difference of two...quantities is equal to the difference of their squares, we have, evidently, =a —b y)=x — y 9 Hence it is obvious that, in these and similar equalities, if...

...extended to quadrinomials — See Euc. П., 4, and Exs. 57, 59, Art. 26. Again, from Ex. 5 we see that the product of the sum and difference of two quantities is equal to the difference of their squares ; and conversely, that the difference of the squares of two quantities is equal to the product of the...

...or a+^b, are called binomial surds, and may be reduced to rational quantities on the principle that the product of the sum and difference of two quantities is equal to the difference of their squares. Thus the binomial surd Multiplied by - —Vab+b Gives a +5, a rational quantity. 312. Trinomial surds...

...2a— 3J. Ans. 4a3-— 9J3. Multiply 3y — c by 3y-\-c. Ans. Qy3 — c3. Thus, by inspection we find, the product of the sum and difference of two quantities is equal to the difference of their squares. The propositions included in this article are proved also in geometry. (Art. 14 ) We can sometimes...