employed three times as a factor, it is raised to the 3d power; if 5 times, to the 5th power, &c. Thus, 2 is the 1st power of 2, or 21. 2× 2 or 4, is the 2d power of 2, or 22. 2×2× 2 or 8, is the 3d power of 2, or 23. 2 × 2 × 2 × 2 × 2 or 32, is the 5th power of 2, or 25. The power is usually denoted by a small figure over the right of the root, called the exponent, or index. When there is no exponent, the number is regarded as the 1st power. The second power is often called the square, because the number of square feet in any square surface, is obtained by multiplying the number of feet in one side by itself. The third power is often called the cube, because the number of cubic feet in any cubical block, may be obtained by raising the number of feet in one side to the 3d power. The 4th power is sometimes called the bi-quadrate, or the square squared; the 5th power, the first sursolid; the 6th power, the square cubed, or the cube squared; the 7th power, the second sursolid; the 8th power, the bi-quadrate squared; the 9th power, the cube cubed; the 10th power, the 1st sursolid squared, &c. If the exponents of any two powers of the same number be added, we shall obtain the exponent of their product. Thus 63 × 65 – 6 × 6 × 6 × 6 × 6 × 6 × 6 × 6 = 68 ; 4x43=4x4x4x4x4–45. = In any two powers of the same number, if we subtract the smaller exponent from the larger, we shall obtain their 6 × 6 × 6 × 6 × 6 × 6 × 6 × 6 quotient. Thus 6865 6×6×6=63. 6 × 6 × 6 × 6 × 6 We may represent any power of a number, by multiplying its exponent. Thus, the 7th power of 5 is 57; the 3d power of 22 is 26, because 22 × 22 × 22-26. properties form the basis of the system of Logarithms. 1. What is the 2d power of 6? the 3d power? 2. Find the value of .94; 123; (†)5; 29. These 5. What is the difference between 34 and 43? 6. What is the value of 117; 37; 23 × 22 ? 7. What power of 9 is equivalent to 95 × 93; 92 × 910; 94 × 96; 9× 97 × 98? 8. Multiply 1279 by 1277, and divide the product by 12715. 9. Divide 319 by 319; 178 by 175; 427 by 426. 10. What is the sixth power of 4? 10 11. What is the 9th power of 53? the 12th power of 185? the 24th power of 172 ? CHAPTER XVI. EVOLUTION. EVOLUTION is the process by which we discover the root of any given power. Thus, 3 is the 2d root of 9, the 3d root of 27, the 5th root of 243, because 9=32, 27=33, 243 35. So the 2d or square root of 49 is 7; the 3d or cube root of 125 is 5; the 4th root of 16 is 2; the 5th root of 1024 is 4, &c. We may denote a root by a radical sign, or by a fractional exponent. The radical sign is √ ́ , and when employed by itself denotes the square root. If we wish to denote the 3d, 5th, 7th, &c. root, the index of the root is written above the radical sign thus, /, /, &c. In fractional exponents, the numerator expresses the power of the number, and the denominator expresses the root. Thus, (27)3 = 3/27; (16) * = √/163 ; (32)3 = &/ 324, &c. The product, or the quotient, of two second, third, or other roots, is the 2d, 3d, &c., root of the product or quotient. Thus, /27x125=27 x 125 or 3375. For 27=33=3× 3 × 3, and 125=5× 5 × 5. Then 27 x 125= 3×3×3× 5x5x5=3×5×3× 5 × 3 × 5=153. Therefore 3/27×125=15. In a similar manner it may be shown that 3375V125/27. The power of any root may be obtained by multiplying the fractional exponent. Thus the 4th power of 273= 8 27. For by the last proposition 272 × 3/272 × 3/272 × 272=278279. The root of any power or root may be obtained by dividing the exponent by the index of the desired root. Thus 3/35-353-315. = 8 1 This is the converse of the last proposition. For if the 3d power of 315 is 31%, or 35, the 3d root of 35 must be 31's. If the numerator and denominator of fractional indices be multiplied or divided by the same number, the value of the quantity is not altered. Thus, 36=312=33. For the multiplication of the numerator involves the number to a certain power, and the multiplication of the denominator extracts the corresponding root. Then the 3d root of the 3d power, the 5th root of the 5th power, &c., is the 1st power. We may multiply or divide any two roots of the same number, by adding or subtracting the fractional exponents. Thus, 2× √2=23+1=26; √5÷÷15—5—4 = 512. For by the last proposition we have 32× √2=√22× or 28. Also 35÷15 23, which is equivalent to √25 or 1 12/54 ÷ 12/53—12/5 or 512. When the exact root of a number can be obtained, it is called a rational number. An irrational number, or surd, is one whose exact root cannot be obtained. Thus, √16, /27, 3/64, 1/81, are rational numbers, equivalent to 4, 3, 4, 3, respectively. But √5, 19, √16, are all surds, and their roots can only be obtained approximately. A number which has a rational root, is called a perfect power. Thus, 16 is a perfect 2d power, and a perfect 4th power, but an imperfect power of any other degree. But 5, 7, 12, &c., are imperfect powers of any degree. 1. What is the square root of 9? the cube root of 8? 2. What is the 4th root of 81? the 5th root of 32? 3. What is the value of I; 25; 64; 6418? 4. What is the product of 8 by 12; 79 by 5. Multiply by 3; 72 by 2/7; 43 1 77? 43 by 42. 6. Divide 63 by 63; 4/5 by 4/5; 173 by 2/17. 7. Find the 4th power of √9; the 6th power of 83. 8. What is the cube root of 76; the 5th root of 11? EXTRACTION OF THE SQUARE ROOT. The square root of a number is the number which, when multiplied by itself, will produce the given number. In the following table are the numbers from 1 to 10 inclusive, and beneath them are their squares; therefore, the numbers of the second line have for their square roots the numbers of the first. Roots 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. If there are decimals in the root, there will be twice as many in the square: because any product contains as many decimals as both factors. And conversely, there will be half as many decimals in the root as in the square. Every entire number, which is not the square of another entire number, is an imperfect second power. For the root of such a number cannot be expressed by a fraction, because a fraction multiplied by itself would give a frac tional product, it must therefore be a surd. Hence we may see that if we divide a square number into periods of two figures, by placing a point over units, and one over each second figure to the left, the number of periods will denote the number of figures in the root. Thus, the square root of 144 contains two figures, and is 12. The square root of 1600 contains 2 figures, and is 40. The square of 30 is 900. The square of 37 is 1369, and to discover in what manner this square is formed, we will multiply 37 by 37, writing each product separately, instead of adding them as we proceed. We then see that we must add the square of the tens, twice the product of the tens by the units, and the square of the units. 37 37 49 21 21 9 1369 1369 (37 9 67) 469 469 Let us now reverse the process, and extract the square root of 1369. Pointing the periods, we find the root will consist of two figures, and the square of the tens must therefore be contained in the 13 hundreds. The greatest square in 13 is 9, and the root 3 is written as the first figure of the required root. Subtracting the square of the root already found, the remainder, 469, must contain twice the product of the tens by the units+the square of the units. To obtain the units' figure, we divide the 46 tens by 2x3=6 tens, which gives a quotient 7. Writing the 7 in the root, and also at the right of the divisor, we multiply by 7, and obtain 469, which is twice the product of the tens by the units plus the square of the units. Hence we deduce the following RULE. Separate the number into periods of two figures each, by placing a point over the units' figure, and another over each second figure to the left (and also to the right, if decimals are desired in the root). Write in the quotient the root of the greatest square contained in the left hand period, and subtract its square from the period. To the remainder annex the second period, and divide the tens of the number thus formed, by twice the first quotient figure, placing the result in the root, and also at the right of the new divisor. Multiply the completed divisor by the new quotient figure, subtract the product from the dividend, and annex the third period to the remainder. Double the root figures already found for a new trial divisor, and proceed as before, until all the periods are brought down. When any trial divisor is not contained in the tens of |