a is called the first power of a, aa the second power, aaa the third power, &c. 10. If a= 2; what will aa be equal to? What will aaa be equal to ? What will aaaa be equal to? 11. What power of a, will a multiplied by aa produce, i. e. what power of a will the first power multiplied by the second. produce? 12. What will represent the products of the following?. Ans. aaa, aa by aaaa аа by aaaaa The product aa may be written a2, aaa, a3, aaaa, a*, &c. The 2, 3, 4 are called indices or exponents. 13. What will represent the product of the following? And what numbers would these products be if a = 2? a by a2 a2 by a2 If we wish to multiply one power of a by another, add together the exponents of the two powers, and the sum will be the exponent of that power of a, which is the product required. 14. It is required to find what power of a, the second pow-er multiplied by the third power will produce. The expo-nents of the two powers are 2 and 3, and their sum is 5. 15. The following products are equal to what powers of a? And equal to what numbers if a=3? a3 X a® a2 X a3 a2 X a3 X at a2 X a® X a5 × a® 16. What is the second power of the third power of a? Ans. a3 X a3 œa®. 17. What is the third power of the second power of a? a2 × a2 × a2 = a® Hence, if we wish to raise any power of a to the second power, multiply its exponent by 2. If we wish to raise it to the third power, multiply its exponent by 3, and if to the fourth, by 4, &c. SECTION LXVIII. On Exponents. 1. What is the quotient of aaa by aa? by aa? aa by a? αααα 2. What is the quotient of a3 by a? a1 by a2? a18 by a12? In order to obtain the quotient of one power of a by another, subtract the exponent of the divisor from the exponent of the dividend. The remainder is the exponent of a in the If we apply this rule to the division of any power of a by the same power, or by a higher power, we obtain results as shown in the following examples: Any power of the product of two or more numbers, is the same as the product of the same powers of those numbers. Thus ab represents the product of a and b, (ab)2 the second power of this product. Then (ab)2= a2 b2 8. If a, b and c are respectively equal to 1, 2 and 3; what will represent the second power of abc? Of a b2c-2? And what numbers will these powers be? 9. What will represent the third powers of a2bc? Of abc? And what numbers will these powers be? SECTION LXIX. On Exponents. We use the fractions,,, for the exponents of a letter, when we wish to signify the,,,and, roots of the number represented by the letter. Thus, at signifies the second or square root of a, 1 the third root of a, at the fourth the fifth root, &c. at xa xat is evidently the third power of the second root of a. Adding the exponents, we have a×a X 3 1 1 at =a, where the denominator shows the root, and the numerator shows the power to which the root is to be raised. Xa=a. Hence aa × aa × a = a+ × a × a xaxat, at x at × at = a‡, aa × aa × aix X at X αἰχ > a = a; therefore a = a, or the third power of the second root of a is the same as the sixth power of the fourth root of a. X 3 6 3 any In like manner it may be shown, that fractional exponent may be changed by multiplying or dividing its numerator and denominator, both by the same number, without altering the value of the number. Thus a3 = a = - a1⁄2 &c. Hence a3× a=a11× aй=at=a× aťì 3 ax 3 And k 2 = 1. Multiply a3 by a3. a3 by at. a3 by a3. 2. Multiply a3bì by a3 b3. a‡ b3 by a3 bì, ab‡ by a3⁄4b. 3. What is the second root of the product of 4 and 9? 4. What is the product of the second roots of 4 and 9? 5. What is the second root of the product of 16 and 25? 6. What is the product of the second roots of 16 and 25? 7. What is the third root of the product of 8 and 27? 8. What is the product of the third roots of 8 and 27? The root of the product of two or more factors, is the same as the product of the roots of those factors. 1 ત 3 Hence (a X a X a)a = aa × xaxa) a2 × a1× a2 = a3 3 may be read the second root of the third power of a, or the third power of the second root of a. As we can raise any power of a to a given power, by multiplying its exponent by the exponent of the power to which it is to be raised, we may obtain the root of any power by dividing the exponent of the power. Thus, the second root of a® is a3, the third root of a is &c. 9. What is the third root of a? 10. What is the sixth root of al2; Of a18? Of a24? 11. What is the second root of a2 b2? Of 4 a2 b2? From this example it is evident, that negative fractional exponents may be used as well as negative integral exponents; and they are to be interpreted the same way. 13. If a=27; what number will a represent? 14. If a=81; what will a represent ? 15. If a = 25 and b=27; what number will present? SECTION LXX. On Logarithms. We give below what is called a table of logarithms. The powers of 2 arranged in columns, and the exponents of those powers placed opposite them on the right. The numbers in the second, fourth and sixth columus, are called the logarithms of those opposite them in the first, third and fifth columns. Whatever power of 2 will make a given number, the exponent of that power is the logarithm of the number in the preceding table. 1. It is required to find the product of 32 by 64. It will be seen by inspecting the preceding table, that 32 is the 5th power of 2, and, that 64 is the sixth; their product, then, is the 11th power of 2. Look in the table amongst the logarithms for 11, and opposite to it stands 2048, which is the number required. 2. Find the product of the following numbers: (2) 16 by 64 32 by 512 (8) 16 by 8192 (10) 256 by 1048576 16384 by 32768 (12) 2048 by 16384 It is required to divide 2097152 by 8192. It will be seen by inspecting the table, that the dividend is the 21st power of 2, and that the divisor is the 13th power of |