Now as the exponents of a and b are equal, we have arrived at the middle of the power, and of course to the highest coefficient. The coefficients now decrease in the reverse order which they increased.

Hence the expanded power is

a*+8a7b+28ab2 +56a5b3+70a4b4+56a3b5+28a2b

+Sab+b2.

Let the reader observe, that the exponent of b, increased by unity is always equal to the number of terms from the beginning, or from the left of the power. Thus, be is in the 3d term, &c. Therefore in finding the coefficients we may divide by the number of terms already written, in place of the exponents of the second term increased by unity.

If the binomial (a+b) becomes (a+1), that is, when b becomes unity, the 8th power becomes,

a*+-8a7+28a+56a5+70a4+56a2+28a2+8a+1. Any power of 1 is 1, and I as a factor never appears. If a becomes 1, then the expanded power becomes, 1+86+2862+5663+7064+5665+286*+862+b2.

The manner of arriving at these results is to represent the unit by a letter, and expand the simple literal terms, and afterwards substitute their values in the result.

(Art. 68.) If we expand (a-b) in place of (a+b), the exponents and coefficients will be precisely the same, but the principles of multiplication of quantities effected by different signs will give the minus sign to the second and to every alternate term.

Thus the 6th power of (a-b) is

a-6a5b+15a4b2---20a3b3+15a3b4-6ab5+6°.

(Art. 69.) This method of readily expanding the powers of a binomial quantity is one application of the "binomial theorem," and it was thus by induction and by observations on the result of particular cases that the theorem was established. Its rigid demonstration is somewhat difficult, but its application is simple and useful.

Its most general form may arise from expanding (a+b). When n=3, we can readily expand it;

When n=4, we can expand it;

When n= any whole positive number, we can expand it. Now let us operate with n just as we would with a known number, and we shall have

(a+b)2= a2+na-b+nan-262,

2

&c

We know not where the series would terminate until we know the value of n. We are convinced of the truth of the result when n represents any positive whole number, but let it be negative or fractional, and we are not so sure of the result. To extend it to such cases requires deeper investigation and rigid demonstration which would not be proper to go into at this time. We shall therefore content ourselves by some of its more simple applications.

EXAMPLES.

1. Required the third power of 3x+2y.

We cannot well expand this by the binomial theorem,

because the terms are not simple literal quantities. But

we can assume

3x=a and 2y=b.

Then

3x+2y=a+b and (a+b)3= a3+3a2b+3ab2+b2.

Now to return to the values of a and b, we have,

Hence (3x+2y3=27x3+54x2y+36xy2+8y3.

2. Required the 4th power of 2a2-3.

Let x=2a2 y=3. Then expand (x-y)4, and return

the values of x and y, and we shall find the result. 16a-96a+216a4-216a2+81.

3. Required the cube of (a+b+c+d).

As we can operate in this summary manner only on binomial quantities, we represent a+b by x, or assume x=a+b, and y=c+d.

Then (x+y)=x3+3x2y+3xy2+y3.

Returning the values of x and y, we have (a+b)3+3(a+b)2(c+d)+3(a+b)(c+d)2+(c+d)3. Now we can expand by the binomial, these quantities contained in parenthesis.

4. Required the 4th power of 2a+3x

Ans. 16a4+96a3x+216a2x2+216ax2+81x4.

5. Expand (x2+3y2)5.

Ans. x10+15xy2 +90x6y4+270x4y6+405x2y +243y10.

6. Expand (2a2+ax)3.

Ans. 8a+12a5x+6a4x2+a3x3.

7. Expand (x-1).

Ans. x-6x5+15x4-20x3+15x2-6x+1. 8. Expand (3x-5)3. Ans. 27x3-135x2+225x-125. 9. Expand (a+2)4. Ans. a*+8a3+24a2+32a+16.

3α2 α3 α
2 216-

11. Expand (a+b+c)2. 12. Expand (a-2b)3. 13. Expand (1-2x)5

EVOLUTION.

CHAPTER II.

(Art. 70.) Evolution is the converse of involution, or the extraction of roots, and the main principle is to observe how powers are formed to be able to trace the operations back. Thus a, to square it we double its exponent, (Art. 65.) and make it a3. Square this and we have a4. Cube a2 and we have a. Take the 4th power of x and we have 24. The nth power of x is x3.

Now, if multiplying exponents raise simple literal quantities to powers, dividing exponents must extract roots. Thus, the square root of a is a2. The cube root of a must be a3. The cube root of a must be its exponent, (1 understood,) divided by 3, or as.

Therefore roots are properly expressed by fractional exponents.

The square root of a is a,

and the exponents,,

1

1

&c. indicate the third, fourth, and fifth roots. The 6th root

5

of 25 is 2; hence we perceive that the numerators of the exponent indicate the power of the quantity, and the denominator the root of that power.

(Art. 71.) The square of ax is a2x2. We square both factors, and so, for any other powers, we raise all the factors to the required power. Conversely, then, we extract roots by taking the required roots of all the factors. Thus the cube root of 8x3 is 2x.

The cube root of the factor 8 is 2, and of the factor x3 is x. The cube root of 16a3 cannot be expressed in a rational quantity, but it can be separated into factors, 8a3 X2, and the cube root of the first factor can be taken, and the index of the root put over the other factor thus, 2ax25, or 2a312. In such cases, the radical sign is usually preferred to the fractional index, as a more distinct separation between the factors.

The square root of 64a is obviously 8a3, and from this and the preceding examples we draw the following

RULE. For the extraction of the roots of monomials. Extract the root of the numeral coefficients and divide the exponent of each letter by the index of the root.

EXAMPLES.

1. What is the square root of 49a2x4?

2. What is the square root of 25c10b2?

3. What is the square root of 20ax?

Ans. Tax2.

Ans. 5c5b.

Ans. 2√5ax.

In 20, the square factor 4 can be taken out, the other factor is 5. The square root of 4 is 2, which is all the root we can take; the root of the other factors can only be indicated as in the answer.

4. What is the square root of 12a2?
5. What is the square root of 144a2c4x2y2?

6. What is the square root of 36x4?

Ans. 2a√3.

Ans. 12ac2xy.

Ans. 6x2.

(Art. 72.) The square root of algebraic quantities may be taken with the double sign, as indicating either plus or minus, for either quantity will give the same square, and we