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Er. 2. Raise 3.7 + 2y to the sixth power.
(54.) By means of the rule just laid down, we are enabled to raise a trinomial or a quadrinomial quantity to any power, without the process of actual multiplication.
Nilus. Suppose it were required to square a + b + c; placing a vinculum over a + T, and viewing it as one quantity, we shall have (a +b+c)' =((a+b) + c)= (a+b)* + 2. (a+b.) ct din a' + 2ab + 1.2 + 2ac + 2lc + d.
Examples. 1. Raise I-a to the 4th power, 2. -a-6 to the 7th power. 3. Img to the gth power. 4. t + a to the 10th power, 5. -2y to the 7th power. 6.
+7 + c to the 2d power. 7. a + b + c to the 3d power. 8. I + y + 3z to the 2d power.
(55.) Evolution, or the extracting of roots, is the reverse of Involution, and is performed by inquiring what quantity, multiplied by itself till the number of factors amounts to the numher of units in the index of the given root, will generate the given quantity.
Ols. By reviewing the tables of roots and their powers in Involution (Art. 48), we find that the cube root of 216 is 6, because 6 multiplied into itself till the number of factors amounts to three, will give 216.-Also,
5 X 5 X 5 X 5 625, :. the 4th root of 625 is 5. rl X-OX-0 = -25 .. the cube root of - 19 is -6.
is the square root of
26 From these data, we discorer that the square, cube, &c. roots of simple and compound quantities, necessarily divide Evolution into several cases.
To extract the roots of simple quantities.
(56.) Rule. Extract the root of the co-efficient for the numerical part, and divide the index of the letter, or letters, by the exponent of ibe proposed root, and it will give the answer required.
Note. Any even ruot of an affirmative quantity may be either + or -, thes the square root of + ais cither + o, or - - a; for (+a)Xitajsta, and (-u)X(-a) - ta?. (Ilins. Art. 29. p.7.)
And an odd root of any quantity will have the same sign as the quantity itself: thus the cube root of 7u3 is +a, and the cube root of
a3 is - a; for (+2)x(+a)x(+a)= tu, and (-u)x(-a)X(-a) --a. (Art. 29.)
Any even root of a negative quantity is impossible; for neither (+a)x(+a for (-a)x(-a) can prodnce -u. (Art. 29.)
The ath root of a product is equal to the nth root of each of the factors multiplied together.
And the nth root of a fraction is equal to the nth root of the numerator divided by the nth root of the denominator.
To find the square root of a compound quantity.
(57) Rule.-Range the quantities according to the dimensions of one of the letters, and set the root of the first term in the quotient.
2. Subtract the square of the root, thus found, froni the firs! term, and bring down the next two terms to the remainder for a dividend.
3. Divide the dividend by double the root, and set the result baik in the quotient and divisor.
4. Multiply the divisor, thus increased, by the term last put in the quotient, and subtract the product from the dividend; and so on, as in common arithmetic.
Examples. | Extract the square rooi of r* - 419 +6r-4r+1.
r* - 4.08 +6r? -4.x + 1(r?-20+1=root
21"-2.x) - 4x + Or
2.ro - 4r+1)2r* - 4+1
2. Extract the square root of 4a' + 12a'r + 13a*r? + bar + ri.
40* +120°r+13a?r? +6ax' + r*(2a? +3ar + r?
4a? + bar +5°) 4a'r? +ar+2*
4a2ro + 6ar! + r*
3. Required the square root of a* + 4aRx+6a?r? + 4ar4 +3.
Ans. a? + 202 + I
Ans. ro -x+1 5. It is required to find the square root of a2 + x2.
gio Ans. at
89 6. Required the square root of 4.x* +6x9 + +3% +156 +25
Ans. 2x +$ 1+5.
To find the roots of powers in general.
(58.) Rule.-1. Find the root of the first term, and place it in the quotient.
2. Subtract its power from that term, and then bring down the second term for a dividend
3. Involve the root, last foun., to the next lowest power, and multiply it by the index of the given power for a divisor.
4. Divide the dividend by this divisor, and the quotient will be the next term of the root
3. Involve now the whole root, and subtract and divide as before ; and so on till the whole is finished.*
Examples. 1. Required the square root of at-2a*: +3a?r?-2ar +..
a-2a’r+3a’rl – 2ax' + x* (a? - artxo
at - 2a’r + a’rl = (as-ar)e
a-2a'r +3a%r? — 2ar + t = (al-ax+x2)
2. Extract the cube root of +6r' _-4079 +963-64.
+6x-40x' +96r-64(x + 2.1-4
3. Required the square root of a’ +2ab +2ac +18 +260 +c.
Ans. a to to. 4. Required the cube root of zo - Or+15.01-20ro +15r5-6r+1.
Ans. x2 - 2x +1. 5. Required the biquadrate root of 16a1-96aor+216aor?—2161 +817".
Ans, 2a-31. 6. Required the fifth root of 32x• ---80x* + 80.r3 — 40.x2 + 100-1.
Ans. 22-1. As this method, in high powers, is generally thought too laborious, it may not be improper to observe, that the roots of compound quantities may sometimes be easily discovered ihus :
1. Extract the roots of some of the most simple terms, and connect them together by the sign + or -, as may be judged most suitable for the purpose.
2. Involve the compound root, thus found, to the proper power, and, if it be the same with the given quantity, it is the root required.
3. But if it be found to differ only in some of the signs, change them from + to -, or from – to +, till its power agrees with the given one throughout.
Thus, in the fifth example, the root 20—3*, is the difference of the roots of the first and last terms; and in the 3d example, the root ato+c is the sum of the roots of the 1st, 4th, and 6th term. The same may also be observed of the 6th example, where the root is found from the first and last terms.
(59.) Equations have received different names, according to the power of the unknown quantity, or quantities, which they contain. Such equations, for example, as contain only the first, or simple power of the unknown quantity, or quantities, are called simPLE EQUATIONS ; those which contain the square, QUADRATIC EQUATIONS ; those which contain the cube, CUBIC EQUATIONS, &c. Thus,
6x +4 = 10r-6} are simple equations ; Or,
10rl + Or = 47-17 Or, 61° +4y= 10y-3)
are quadratic equations ; 10r-2r=6r+17 Or, roy+ yo=4r. --Gr are cubic equations. -3b8. There are four processes, or rules, which are to be applied in the same way, and in the same order, for the resolution of all equations, and which will, as it were mechanically, determine the value of the unknown quantity. These processes are grounded upon the following
AXIOMS. (60.) 1. If equal quantities be added to equal quantities, their sums will be equal.
2. If equal quantities he subtracted from equal quantities, their remainders will be equal.
3. If equal quantities be multiplied by equal quantities, their products will be equal.
4. If equal quantities be divided by equal quantities, their quotients will be equal.
(61.) Rule I. Any quantity may be transferred from one side of the equation to the other, by changing its sign.
Notc.- This rule is founded on Axioms I. and II.
EXAMPLE I. If x + 8= 15, by subtracting 8 from each side of the equation, then will I + 8 – 8 - 15 8 — 8 = 0; therefore r = 15
- 8. COROL.-Hence it appear's, 'that + 8 may be transfrrred to the other side of the cqnation by changing it to - 8 ; and we obtain, by this means, the value of x, which is 15 8. or 7.
2. If x + 3 = 7, tlien will r = 7 34.
d, then will r =(-d + Fl. 4. If 41 8= 3.x + 20, then will 41 3.7 = 20 + 8, of I 28.