Resolve into prime factors with fractional exponents: 65. 12, 72, 96, V64; and find their product. 82. A vessel that has two pipes can be filled in 2 hours less time by one than by the other, and by both together in 1 hour 52 minutes 30 seconds. How long will it take each pipe alone to fill the vessel ? 83. A number is expressed by two digits, the second of which is the square of the other, and when 54 is added its digits are interchanged. Find the number. 84. Divide 35 into two parts such that the sum of the two fractions formed by dividing each part by the other may be 2. 85. A number consists of three digits in arithmetical progression; and this number divided by the sum of its digits is equal to 26; but if 198 is added to the number, the digits in the units' and hundreds' places will be interchanged. Find the number. 86. The sum of the squares of the extremes of four numbers in arithmetical progression is 200, and the sum of the squares of the means is 136. What are the numbers? 87. Show that if any even number of terms of the series 1, 3, 5 is taken, the sum of the first half is to the sum of the second half in the ratio 1 : 3. 97. If a, b, c, d are in continued proportion, prove that b+c is a mean proportional between a + b and c + d. 98. If a b:b+c=c+d:d+a, prove that a = c, or a + b + c + d = 0. 99. The number of eggs which can be bought for 1 dollar is equal to twice the number of cents which 32 eggs cost. How many eggs can be bought for 1 dollar? 100. Find two fractions whose sum is, and whose difference is equal to their product. 101. The velocity of a falling body varies as the time during which it has fallen from rest, and the velocity at the end of 2 seconds is 64 ft. Find the velocity at the end of 6 seconds. 102. The distance through which a body falls from rest varies as the square of the time it falls; and a body falls 144 ft. in 3 seconds. How far does it fall in 4 seconds? 103. The volume of a gas varies directly as the absolute temperature and inversely as the pressure. If the volume of a gas is 1 cubic foot, when the pressure is 15 and the temperature 280, what will be the volume when the pressure is 35 and the temperature 320 ? 104. The difference between the first and second of four numbers in geometrical progression is 96; the difference between the third and fourth is 6. Find the numbers. 105. If a2, b2, c2 are in arithmetical progression, prove that b+c, c+a, a + b are in harmonical progression. When x = ∞, and when x = 0, find the limit of: Resolve into factors and find all the values of x: 129. √5x+1+2√4x-3= 10 √x 2. Expand: 132. (a1 — x3)o; (3a3 — 2 b2)5; (a3 — ‡ b1)7; $ (2 − ‡ x2)o; (3 − ‡ x*)7; (x2 — x−2)3. 133. (x2 — y3)5; (2x2 +√3 x)1; (√a2+1+a3)3; (1 + 2 x - x2 - x3)3. 134. Expand to four terms (1 − 3 x) ̄*; (1 − 4 x2) ̄*; (1 − § x*)*; (a − 2 x ̄†) – 3. 135. Find the eighty-seventh term of (2 x − y)9o. CHAPTER XXVI. LOGARITHMS. 411. If numbers are regarded as powers of ten, the exponents of the powers are the Common or Briggs Logarithms of the numbers. If A and B denote two numbers, a and b their logarithms, then 10" A, 10' B; or, written in logarithmic = form, log A = a, log B = b. 412. The logarithm of a product is found by adding the logarithms of its factors. For AX B = 10a × 10b = 10a+b. (§ 244) Therefore, log (A x B) = a + blog A+ log B. 413. The logarithm of a quotient is found by subtracting the logarithm of the divisor from that of the dividend. 414. The logarithm of a power of a number is found by multiplying the logarithm of the number by the exponent of the power. |