...power of a quantity, it always means the power of its numeral value considered abstractly. (ART. 17.) THE PRODUCT OF THE SUM AND DIFFERENCE OF TWO QUANTITIES IS EQUAL TO THE DIFFERENCE OF THEIR SQUARES, as will be seen by inspecting the following products : The first example should be multiplied in full...

...a be one of the quantities and b the other ; then (a + b) (a — b) = a? — b2. We thus learn that the product of the sum and difference of two quantities is equal to the difference of their squareS. These propositions are demonstrated, in geometry, in another form, where it is shown, 1°, That if...

...^ But a+6 represents the sum of two quantities, and a — b, their difference ; hence, THEOREM III. The product of the sum and difference of two quantities, is equal to the difference of their squares. EXAMPLES. 1. (5+3)(5-3)=25-9=16=8X2. 2. (2a+6)(2a— 6)=4a*— V. 3. (2z+3y)(2x-3y)=4z'-9i/2. 4. (5a+46)(5a—...

...j02=4z2— 4ijr+y2. 3. (3x—5zy=9xi— 30xz+25z2. 4. (02— 3cx)2=o2z2— 6 ART. 8O. THEOREM III. — The product of the sum and difference of two quantities, is equal to the difference of tlieir squares. Let a represent one of the quantities, and 6 the other ; then o-|-i=their sum, and...

...+6 By a— b -ab—b2 Product a2— 62 The following formula is obtained from this example : — " the product of the sum and difference of two quantities, is equal to the difference of their squares." And, for the reason just given [35] it, also, is general. 37. EXAMPLE 3. — Multiply c — d By c...

...5— x/3 Product, 25-3=22, as required. These two examples are comprehended under the Rule in Art. 62, the product of the sum and difference of two quantities is equal to the difference of their squares. Ex. 3. Find a multiplier that shall make v/5+v/3 rational and determine the product. Ex. 4. Find a...

...number of factors. The following are some of the properties of products with respect to their forms : 1. The product of the sum and difference of two quantities is equal to the difference of their squares ; that is, (x + y) (x — y) = x' — y'. 2. Twice the sum of two squares is equal to the sum of two...

...quantity, minus the double product of the first by the second, plus the second power of the second. 5. The product of the sum and difference of two quantities is equal to the difference of their second powers. The questions, art. 15, will furnish additional exercises for the learner in stating...

...beginners often commit the mistake of putting the square of a— b equal to af—h\ THEOREM III. (67.) The product of the sum and difference of two quantities is equal to the difference of their squares. Thus, if we multiply a +b by a —b a'+ab -ab-b' we obtain the product a'- b'. Examples. Ex. 1. (3a+^b)(3a-4b)=9a'-16b\...

...following manner. It is easily seen by actual multiplication, that (a + 6) x (a - 6) = a- - b\ or that the product of the sum and difference of two quantities is equal to the difference of their squares; a theorem which may be used in the multiplication of such quantities as those in the last example,...