« ΠροηγούμενηΣυνέχεια »
p' which is the second power of L. Therefore the greatest
possible product is when the two parts are equal.
In the above example 1 = 8, and are
= 64. This is the
4 greatest possible product that can be formed of two numbers whose sum is 16. It was therefore absurd to require the product to be 72; and the imaginary values of x arise from that absurdity.
2. It is required to find a number such, that if to its second power, 9 times itself be added, the sum will be equal to three times the number less 5.
** + 9x= 3x - 5.
are + 6 x = -5. This equation is in the form of 2 + px = -9, which gives
2 Putting in the values of p and q *=-3+ (9-5)*
- 3 + 2. The values are -1 and — 5, both negative. Consequently neither value will answer the conditions of the question. This shows also that those conditions cannot be answered.
But if we change the sign of x in the equation, that is, put in x instead of x, it becomes x2 9x= 3 x
- 5. Changing all the signs
9 x = 3x + 5. This shows that the question should be expressed thus :
It is required to find a number, such, that if from 9 times itself, its second power be subtracted, the remainder will be equal to 3 times the number plus 5.
The values will both be positive in this, and both answer the conditions.
* = 3 + (9—5)* = 3 +2. The values are 5 and 1 as before, but now both are positive, and both answer the conditions of the question.
3. There are two numbers whose sum is a, and the sum of whose second powers is b. It is required to find the numbers.
Examine the various cases which arise from giving different values to a and b. Also how the negative value is to be interpreted. Do the same with the following examples.
4. There are two numbers whose difference is d, and the sum of whose second powers is b. Required the numbers.
5. There are two numbers whose difference is a, and the difference of whose third powers is b. Required the numbers.
6. A man bought a number of sheep for a number a of dollars; and on counting them he found that if there had been a number b more of them, the price.of each would have been less by a sum c. How many did he buy?
7. A grazier bought as many sheep as cost him a sum a, out of which he reserved a number b, and sold the remainder for a sum c, gaining a sum d per head by them. How many sheep did he buy, and what was the price of each?
8. A merchant sold a quantity of brandy for a sum a, and gained as much per cent. as the brandy cost him. What was the price of the brandy?
XXXVI. Of Powers and Roots in General.
Some explanation of powers both of numeral and literal quantities was given Art. X. The method of finding the roots of the second and third powers, that is, of finding the second and third roots of numeral quantities, has also been explained ; and their application to the solution of equations. But it is
frequently necessary to find the roots of other powers, as well as of the second and third, and of literal, as well as of numeral quantities. Preparatory to this, it is necessary to attend a little more particularly to the formation of powers.
The second power of a is a x a= a*.
If a quantity as a is multiplied into itself until it enters m times as a factor, it is said to be raised to the mth power, and is expressed a". This is done by m-1 multiplications; for one multiplication as a x a produces a’ the second power, two multiplications produce the third power, &c.
We have seen above Art. X. that when the quantities to be multiplied are alike, the multiplication is performed by adding the exponents. By this principle it is easy to find any power of a quantity which is already a power. Thus
The second power of a' is a' X a = 224% = a®.
+2+ = a®. The second power of am is am xam = amtm - o oam. The third power of an is am xam xam = amt.mtm = a3m.
The mth power of a?is ax al x al x a' x ... =a2+3+3+2+......, until a' is taken m times as a factor, that is, until the exponent 2 has been taken m' times. Hence it is expressed aRm.
The nth power of am is am xam xam until m is taken n times, and the power is expressed amn.
N. B. The dots ..... in the two last examples are used to express the continuation of the multiplication or addition, because it cannot come to an end until m in the first case, and n in the second, receive a determinate value.
In looking over the above examples we observe ;
1st. That the second power of a is the same as the third power of a', and so of all others.
2. That in finding a power of a letter the exponent is added until it is taken as many times as there are units in the expo nent of the required power. Hence any quantity may be raised to any power by multiplying its exponent by the exponent of the power to which it is to be raised.
The 5th power of a® is qox6 = al.
The power of a product is the same as the product of that power of all its factors.
The 2d power of 3 ab is 3 ab x 3 ab = 9 a*b*.
Hence, when a quantity consists of several letters, it may be raised to any power by multiplying the exponents of each letter by the exponent of the power required ; and if the quantity has a numeral coefficient, that must be raised to the power required.
The powers of a fraction are found by raising both numerator and denominator to the power required; for that is equivalent to the continued multiplication of the fraction by itself. 1 What is the 5th power of 3 a' b3 m?
2 ac 2 What is the 3d
Powers of compound quantities are found like those of simple quantities, by the continued multiplication of the quantity into itself. The second power is found by multiplying the quantity once by itself. The third power is found by two multiplications, &c.
The powers of compound quantities are expressed by enclosing the quantities in a parenthesis, or by drawing a vinculum over them, and giving them the exponent of the power. The third power of a + 2 b - c is expressed (a + 2b-c)'; or
a + 2b
The powers are found by multiplication as follows:
a + 2b-C a + 2b-C
a* + 2 ab-ac
- ac – 26c+ c
a' + 4 a b +4 ab — 2 ac-4abc+ac
- ac – 4abc-46*c+2ac +46c-*
ad + 6 a' b + 12 a 2 + 863 -3 ac 12 abc-12 b c
+ 3 ad + 6 b c — C= (a +26—c)'. If the third power be multiplied by a + 2b-c, it will produce the fourth power.
3. What is the second power of 3c +2d?
4. What is the third power of 4 a = bc?
5. What is the fifth power of a-b?
6. What is the fourth power of 2 a'c-r?
In practice it is generally more convenient to express the powers of compound quantities, than actually to find them by multiplication. And operations may frequently be more easily performed on them when they are only expressed.
(a + b) x (a + b) = (a + b)242 = (a + b)"