An Algebra Upon the Inductive Method of Instruction

This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1840 edition. Excerpt: ... X aT X ax = a; therefore a2 = ar, or the third power of the second root of a is the same as the sixth power of the fourth root of a. In like manner it may be shown, that any fractional exponent may be changed by multiplying or dividing its numerator and denominator, both by the same number, without altering 3 4 8 the value of the number. Thus a2 = a = aTi &c. Hence a? X a = oT5 X oTT = cft2=aX a" And a X a = a X a = a a i 2 l i I 1. Multiply a2 by a1, a by aT. aT by a. II 31 11 11 I 1 2. Multiply a b3 by a2 6. a5 ft3" by a2 6T. a67 by aT6. 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. Hence (aX aX a) = a? X a X a = a 3 a82 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 a8 is a3, the third root of a6 is a2, &c. 9. What is the third root of a9? 10. What is the sixth root of ai2; Of a18? Of a24? 11. What is the second root of a2 b? Of 4 a2 b2? i I 12. Divide a3 by a3. Subtracting their exponents, we have But _! 1 Hence a 3 =--a3 From this example it is evident, that negative fractional exponents may be used as well as negative integral exponents; and...