CHAPTER VII. POWERS. SECTION I. Involution of Simple Quantities. 1. WHAT is the product of a a multiplied by a? ANS. u3. The answer to this question, obtained by the common process of multiplication, is a a a. Instead of repeating the letter, we indicate the number of times it occurs in the product, by the small figure placed above it on the right hand. This figure is called its Exponent. It shows the Power of the letter; that is, how many times it is used as a factor in the multiplication. When a letter has no number annexed, the exponent is always a unit, or 1, and the letter is said to be the first power or Root; thus, a Xa a xa x a = a xa xa xa = a is the root, or first power; a2 is the second power; a3 is the third power; a1 is the fourth power, &c. The second power is sometimes called the square; the third power, the cube; and the fourth power, the biquadrate. If we suppose the value of a to be 3, α= 3, the first power; a2 = 32, or 3 x3 = a3 33, or 3 x 3 x3 = a4=34, or 3 × 3 × 3 × 3 = 9, the second power; 27, the third power; 81, the fourth power, &c. The coefficient and the exponent of a quantity, being very different things, must not be confounded together. Thus, the values of 5 x and x5 are far from being equal. Let the value of x be 6; then 5 x 5 x 6 = 30; but x56 × 6 × 6 × 6 × 6 = 7776. It is evident that we raise a single letter to any prɔposed power by giving it the exponent of that power. 2. What is the sixth power of x? the fourth power of d? the fifth power of c? the eighth power of a? the seventh power of x? 3. What is the product of a b multiplied by a b; that is, what is the second power of a b? a b xa ba a b b, or a2 b2. Here, the quantity to be raised consists of two factors, a and b; and the required power is expressed by the same factors, with the exponent of that power written above each. 4. What is the fourth power of x y? 5. Raise a b c to the sixth power. 6. What is the ninth power of a b x? 7. Involve m n y to the seventh power. 8. What is the square of 4 a b? It will be remembered, that the square, or second power of any quantity, is the product of that quantity ANS. 16 a2 b2. multiplied by itself; consequently, the second power of 4 a b is 4 a b × 4 a b = 16 a a b b, or 16 a2 b2. Hence, coefficients must be raised to any required power, by actual multiplication. 9. What is the third power of 5 a x? 10. What is the fourth power of 7 a b c? 11. Raise 6 x y z to the fifth power. 12. What is the fourth power of — 2 a b c? c, first power. 2 a b c +16 a4 b4c4, fourth power. Hence it appears, that when the root or first power is a negative quantity, the ODD powers are negative, and the EVEN powers are positive. 13. What is the second power of a3? ANS. a6. Here, the quantity to be involved is already a power, and the exponent is multiplied by the exponent of the power proposed; thus, a3×2 = a; for a3 = a a a, and a a a Xa a a = ao. ANs. a3 b9. 14. What is the third power of a b3? SECTION II. Involution of Compound Quantities. 1. It is required to find the fifth power of the binomial quantity a+b. b2 a3+3a2b+3 a 62 + 63, third power. a + b a2 + 3 a3 b+3a2 b2 + a b3 a3 b + 3 a2 b2 + 3 a b3 + b2 at +4 a3 b + 6 a2 b2 + 4 a b3 +b4, fourth power. a + b a3 + 4 aa b + 6 a3 b2 + 4 a2 b3 + a b1 a1 b + 4 a3 b2 + 6 a2 b3 + 4 a ba + b5 a5+5a4b+10 a3 b2+10 a2 b3 +5 a b + b5, ANS. The powers of compound quantities are obtained by actual multiplication. These powers, however, are often expressed by means of an exponent. Thus, (a+b+c)3 indicates the third power of a + b + c |