SECTION X. INVOLUTION AND POWERS. (118.) INVOLUTION is the process of raising a quantity to any proposed power. The products formed by the successive multiplication of the same number by itself are called the powers of that number. See Art. 26. Thus The first power of 3 is 3. The square, or second power of 3, is 9, or 3×3. The cube, or third power of 3, is 27, or 3×3×3. The fourth power of 3 is 81, or 3×3×3×3, etc., etc., etc. (119.) The exponent is a number or letter written a little above a quantity to the right, and denotes the number of times that quantity enters as a factor into a product. See Art. 27. Thus the first power of a is a', where the exponent is 1, which, however, is commonly written a, the exponent being understood. The second power of a is aXa, or a', where the exponent 2 denotes that a is taken twice as a factor to produce the power aa. The third power of a is a×a×a, or a3, where the QUEST. Give the definition of the word power. Define the term exponent. exponent 3 denotes that a is taken three times as a factor to produce the power aaa. The fourth power of a is a×a×a×a, or a1, etc. PROBLEM I. (120.) To square a monomial. Let it be required to raise the monomial 3ab' to the second power. According to the rule for the multiplication of monomials, Arts. 55 and 56, (3ab1)2=3ab1×3ab2=9a2b1. So, also, (4a2bc3)2=4a2bc3×4a2bc3=16a*b*c®. Hence, in order to square a monomial, we have the following RULE. Square the coefficient and multiply the exponent of each of the letters by 2. 4. What is the square of 12a'xy? 5. What is the square of 15ab3cx*? 6. What is the square of 18x'yz'? Ans. 64a*b*x*. Ans. Ans. Ans. (121.) We have seen in Art. 62 that + multiplied QUEST.-Explain the method of squaring a monomial. Give the rule for squaring a monomial. by +, and multiplied by give +. Now the square of any quantity being the product of that quan tity by itself, it follows that Whatever may be the sign of a monomial, its square will be positive. 7. What is the square of -10ab3x* ? Ans. +100a*b*x® (122.) To raise a monomial to any power. Let it be required to raise the monomial 2ab3 to the fifth power. According to the rules for multiplication, =32a1o15; where we perceive, 1st. That the coefficient 2 has been raised to the fifth power. 2d. That the exponent of each of the letters has been multiplied by 5. In like manner, (3a2b3c)3=3a2b3c×3a2b3c×3a2bc where we perceive that the coefficient 3 has been rais QUEST.-What will be the sign of the square? How do we raise monomial to any power? ed to the third power, and that the exponent of each of the letters has been multiplied by 3. Hence, to raise a monomial to any power, we have the following RULE. 1. Raise the numerical coefficient to the given power by actual multiplication. 2. Multiply the exponent of each of the letters by the exponent of the power required. Examples. 1. What is the cube of 6xy*z*? Ans. 216x3y 212 2. What is the fourth power of 4ab3c3 ? Ans. 256a'b'c12? 3. What is the fifth power of 3ax'y'? (123.) We have seen that, whatever may be the sign of a monomial, its square is always positive. It is obvious that the product of several factors which are all positive is invariably positive. Also, the product of an even number of negative factors is positive, but the product of an odd number of negative factors is negative. Thus QUEST.-Give the rule for raising a monomial to any power? How do we determine the sign of the power? -ax-a=+a"; -ax-ax-a=— a3 ; -ax-ax-ax−a=+a*; -ax-ax-ax-ax-a-ao, etc., etc., etc. Hence we perceive that Every EVEN power is positive, but an ODD power the same sign as its root. 7. What is the cube of -2ac3x*? has Ans. -8a'c'x15. 8. What is the fourth power of —7ab3ď3y' ? 9. What is the fifth power of 2c3mx2? Ans. +32cm3x1o. 10. What is the fifth power of 3bc'dy"? 11. What is the fourth power of -4ad*x*? Ans. +256aˆd2x2o. 12. What is the cube of -8acx'y'? Ans. -512a'c'x'y'. 13. What is the cube of -4ab❜my'? Ans. -64a b'm3y°. 14. What is the fifth power of —2bc3d3x3 ? Ans. 15. What is the fourth power of 5c'x'y'? QUEST.-What is the rule for the sign of the power? |