To involve, or raise, nurd quantities to any power. BULE. Multiply the index of the quantity by the index of the power, to which it is to be raised, and annex the result to the power of the rational parts, and it will give the power required. EXAMPLES. 1. It is required to find the square of a3. 2 of gas. Therefore 3«3|2 =÷.a+\3=-÷3 /a3, the square required. 2. It is required to find the cube of $7. 3 Therefore 7.73-124. 343, the cube required. 3. Required the square of 33. Ans. 99 EXAMPLES. 1. Required to find the product of 3✅8 and 2✅6. Here, 3×2×8×√6=6√8x6=6√48=616x3=6x4x √3=243, the product required. 2. Required to find the product of 3 and 23 √5. 3 3 号 10 3 3 =√15=√15, the product required. 3. Required the product of 58 and 35. Ans. 30/10. 4. Required the product of 136 and 33✓18. 5. Required the product of ✓ and √7%. Ans. 3√4. 1. Reduce the surds to the same index. 2. Then take the quotient of the rational quantities, and to it annex the quotient of the surds, and it will give the whole quotient required; which must be reduced to its most simple terms. EXAMPLES. 1. It is required to divide 8108 by 2✔6. 8÷2x108-6=4√18=4√9x2=4x3√2=12√2 the quo tient required. 2. It is required to divide 83✓512 by 43√2. To involve, or raise, surd quantities to any power. RULE. Multiply the index of the quantity by the index of the power, to which it is to be raised, and annex the result ta the power of the rational parts, and it will give the power required. EXAMPLES. 1. It is required to find the square of a3. 2 Therefore a3 3 a3, the square required. 2. It is required to find the cube of PV7. 4. Required the cube of 2^, or ✔2. 5. Required the 4th power of √6. 6. It is required to find the nth power of a. Ans. 2/2. Ans. Ans. PROBLEM IX. To extract the roots of surd quantities. RULE.* Divide the index of the given quantity by the index of the root to be extracted; then annex the result to the root of the rational part, and it will give the root required. EXAMPLES. I. It is required to find the square root of 93√3. First, 9=3; And 332=33÷2_3} ; Therefore 93 =3×3 is the square root required. 2. It is required to find the cube root of √2. First 3✓=; * The square root of a binomial or residual surd, A+B, or A-B, may be found thus: take ✔ — B2=D; Thus, the square root of 8+2√7=1+√7; And the square root of 3—√8=√2—1 ; But for the cube, or any higher root, no general rule is given. Thereforc }√2|3 =1×2 is the cube root required. 3. Required the square root of 103. 4. Required the cube root of a3. 5. Required the 4th root of 3x2. Ans. 10/10. Ans. 3xx. INFINITE SERIES. An Infinite Series is formed from a fraction, having a compound denominator, or by extracting the root of a surd quantity; and is such as, being continued, would run on infinitely, in the manner of some decimal fractions. But by obtaining a few of the first terms, the law of the progression will be manifest, so that the series may be continued without the continuance of the operation, by which the first tjrms are found. PROBLEM i. To reduce fractional quantities to infinite series. RULE. Divide the numerator by the denominator; and the operation, continued as far as may be thought necessary, will give the series required. 1—x) 1 (l+x+x2+x3+x*+, &c.=; |