mainder will be less than 113. Required the price of the watch. 6. What number is that whose half and third part added together are less than 105, but its half diminished by its fifth part is greater than 33? 7. The double of a number diminished by 6 is greater than 24, and triple the number diminished by 6 is less than double the number increased by 10. Required the number. SECTION X. INVOLUTION AND POWERS. (136.) According to Art. 20, the products formed by the sucressive multiplication of the same number by itself are called the powers of that number. Thus, the first power of 3 is 3. The second power of 3 is 9, or 3×3. The fourth power of 3 is 81, or 3×3×3×3, According to Art. 21, 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. Thus, the first power of a is a', where the exponent is 1, which, however, is commonly omitted. The second power of a is a×a, or a2, where the exponent 2 denotes that a is taken twice as a factor to produce the pow er aa. The third power of a is a×a×a, or a3, where the exponent denotes that a is taken three times as a factor to produce he power aaa. The fourth power of a is a×a×a×a, or a1. sa factor n times, and is written a". repeated Exponents may be applied to polynomials as well as to moDomials. Thus (a+b+c) is the same as (a+b+c)x(a+b+c)×(a+b+c), or the third power of the entire expression a+b+c. (137.) According to the rule for the multiplication of mono mials, Arts. 49 and 50. So, also, (3ab')'=3ab'x3ab2=9a'b'. (4abc")'=4abc" x 4a2bc16a'b'c'. Hence it appears that, in order to square a monomial, we must square its coefficient, and multiply the exponent of each of the letters by 2. EXAMPLES. 1. Required the square of Taxy. Ans. 49a2x2y3. 2. Required the square of 11abcd. 3. Required the square of 12a'xy. 4. Required the square of 15ab'cx*. 5. Required the square of 18.xyz". According to Art. 53, + multiplied by +, and — multiplied by -, give +. Now the square of any quantity being the product of that quantity by itself, it necessarily follows that whatever may be the sign of a monomial, its square must be affected with the sign +. Thus the square of +3ax or of −3ax is +9a2x2. (138.) The method of involving a quantity to any power, is easily derived from the preceding principles. Let it be required to form the fifth power of 2a3b3. (2a'b')'=2a'b1×2a3b3×2a3b3×2a3b3×2a*b2 Where we perceive 1. That the coefficient has been raised to the fifth power. 2. That the exponent of each of the letters has been multiplied by 5. In like manner, (3a'b'c)'=3a'b'c3a b'c × 3a b'c =27a b'c3. Hence, to raise a monomial to any power, we have the following RULE. Raise the numerical coefficient to the given power, and multiply the exponent of each of the letters by the exponent of the power required. EXAMPLES. 1. Required the fourth power of 4ab3c3. 2. Required the fifth power of 3ax'y'. Ans. 256a b c1. (139.) Let us now consider the sign with which the power should be affected. We have seen, Art. 137, that whatever may be the sign of a monomial, its square is always positive. It is obvious, from the same considerations, that the product of an even number of negative factors is positive, but the product of an odd number of negative factors is negative Thus, -a×a=+a3 -ax-ax-a=—a3 -ax-ax-ax—a=+a* -ax-ax-ax-ax-a=—a &c., &c., &c. The product of several factors which are all positive, is in variably positive. Hence, Every EVEN power is positive, but an ODD power has the same sign as its root. EXAMPLES. 1. Required the square of -2x3. 2. Required the square of -3x". 3. Required the cube of —3a3. 4. Required the fourth power of —3a3b3h. Ans. +4x1o. (140.) A fraction is involved by involving both the numerator (141.) Hence, expressions with negative exponents ar volved by the same rule as those with positive exponents. Thus, let it be required to find the square of a3. 1 This expression may be written which, raised to the or a, the same result as would be obtained by multiplying the exponent -3 by 2. (142.) A polynomial is involved by multiplying it into itself as many times less one as is denoted by the exponent of the power. Ex. 1. Required the fourth power of a+b. a + b a + b a2+ab +ab+b2 (a+b)2=a2+2ab+63, the second power of a+b. a+b a3+2ab+ab3 +ab+2ab+b* (a+b)3=a3+3a2b+3ab2+b3, the third power. a + b a*+3a3b+3a2b2+ ab3 +ab+3a'b'+3ab'+b* (a+b)*=a*+4a3b+6a2b2+4ab3+b', the fourth power. Ex. 2. Required the fourth power of a-b. Ans. a*-4a3b+6a2b3—4ab3+b* Ex. 3. Required the cube of 2a-1. |