27. In a mixture of 60 gallons, there are 4 parts of alcohol, 8 parts of water, and 3 parts of vinegar; how many gallons of each in the mixture? 28. An estate of $10000 was divided between two sons in the ratio of to ; what was the share of each? 29. Three schools containing 30, 40, and 50 pupils, respectively, are required to furnish $60 to purchase a library; what is the proportionate share of each school? 30. A quantity of gold was mixed in the proportion of 5 pwt. of 20 carats, and 6 pwt. of 18 carats; how much of each in 22 ounces of the mixture? INVOLUTION. Involution is the method of finding any power of a number. A Power of a number is the number itself, or the product obtained by using the number two or more times as a factor. A Root of a number is the number itself, or one of the equal factors by whose product the number is produced. Thus, 8 is the third power of 2, and 2 is the third or cube root of 8.. The Exponent of a power is a number placed at the right of the root and just above it, to show the number of times the root is to be used as a factor. It also denotes the degree of the power. Thus, 22 denotes the second power of 2, or 2 × 2; 23 = 2 × 2 × 2, or the third power of 2. The Square of a number, or its second power, is so called because the area of a square is obtained by the product of two equal factors, each of which represents the length of one of the sides of the square. 26 92 THE NEW AMERICAN ARITHM The Cube of a number, or its third power, is so called because the solidity of a cube is obtained by the product of three equal factors, each of which represents the length of one of the sides of the cube. A Parenthesis () is used to include such numbers as are to be considered together. A Vinculum is sometimes used for the same purpose. Thus, (10+ 5) x (10+ 5), or 10+5 x 10+5 indicates that the sum of 10+ 5 is to be multiplied by the sum of 10+ 5, or 15 × 15 = 225. ORAL EXERCISES. 1. What is the product of 2 used 3 times as a factor, or 23? What is the square of 6? 2. What is the third power of 3? 3. What is the cube of 4? 6. What is the cube of? 7. The side of a square is 7 feet; how many square feet does it contain? 8. A box measures 5 feet in length, 5 in breadth, and 5 in depth: how many cubic feet does it contain? 9. What is the fourth power of 3? 10. Raise to the fourth power. WRITTEN EXERCISES. 1. What is the third power of 25? ANALYSIS.-We write 25 three times as a factor, and perform the multiplication. Thus, 25 × 25 × 25 — 625, the third power of 25, or 253. = RULE. Write the given number as many times as a factor as there are units in the exponent, or number expressing the required power; the product of these factors will be the required power. 2. Raise 36 to the fifth power. 3. What is the value of 65? 7. Raise 21 to the fourth power. 8. What is the value of 32 × 33? ANALYSIS.-32 = 3 × 3, and 333 × 3 × 3; 32 × 33 = 3x 3 × 3 × 3 × 3 = 35 = 243. Note.-As an exponent indicates a number of factors, it is evident that in the product of any number of powers of the same number there will be as many factors as there are units in the sum of the exponents; we can therefore represent the product of such powers by adding the exponents. Thus, 32 × 33 = 32 + 3 = 35. : = 9. 62 × 63 × 6 × 65 what power of 6? 10. What is the value of 44 × 42 × 43? INVOLUTION BY ANALYSIS. 1. Raise 15 to the second power by multiplying its tens and units separately. ANALYSIS.-152 (10+ 5) x (10+5). = 10 x 10 = 100 = the square of the tens. 10 x 5 = 5 X 5 = 50} = twice the product of the tens by the units. = 25 = the square of the units. 225 This analysis indicates that the square of a number consisting of tens and units is equal to the square of the tens, plus twice the product of the tens by the units, plus the square of the units. To. prove the accuracy of the proposition, let the tens of a number be denoted by a, and the units by b, then the square of the number will be denoted by (a+b) × (a+b) = a2+2 ab+b2; in which result a2 the square of the tens, 2ab= = twice the product of the tens by the units, and b2= the square of the units. The inference drawn from the arithmetical analysis is therefore correct. = 2. Raise 15 to the third power by multiplying its tens and units separately. = = 1500=3 times the square of the tens multiplied by the units. 7503 times the square of the units multiplied by the tens. = 125 the cube of the units. = 3375 (10+5) × (10+5) × (10+5)=153. = In this operation we see that the cube of a number consisting of tens and units is equal to the cube of the tens, plus three times the square of the tens multiplied by the units, plus three times the square of the units multiplied by the tens, plus the cube of the units. To prove that this result is general, let a denote the tens and b the units of any number, then the cube of the number will be denoted by (a+b) x (a + b) × (a+b) = a3 +3 a2b+3 ab2+b3; in which result as the cube of the tens, 3a2b= three times the square of the tens multiplied by the units, 3ab2=three times the square of the units multiplied by the tens, and b3— the cube of the units. The principle stated is therefore general in its application. = = 3. Raise the following numbers to the required powers by multiplying their tens and units separately: 252, 552, 652, 992, 253, 453, 553. EVOLUTION. Evolution is the method of finding the root of any given power. Roots are indicated by writing the Radical Sign / before the number. Thus, 19 denotes the square root of 9. The Index of the root is a small figure placed over the radical sign to indicate the degree of the root. Thus, 27 denotes the third or cube root of 27. In the square root the index 2 is usually omitted. Roots are sometimes indicated by a fractional exponent. Thus, 9* denotes the square root of 9. The Square Root of a number is one of the two equal factors by whose product the number is produced. The Cube Root of a number is one of the three equal factors by whose product the number is produced. Numbers that have exact roots are called Perfect Powers; all others are termed Imperfect Powers. Roots that can be found exactly are called Rational Roots; all others are called Irrational or Surd Roots. ORAL EXERCISES. 1. What is one of the two equal factors by whose product 9 is produced? 2. Name one of the two equal factors of 16. 3. Name one of the three equal factors of 8. 4. What is the square root of 25? 5. What is the square root of 36? 6. What is the cube root of 27? 7. What is the third root of 64? 8. The equal factors of 15625 are 25, 25, and 25; what root of 15625 is 25? |