Imágenes de páginas
PDF
EPUB

For,

(ab)" =(ab) (ab) (ab)

...

• to n factors

= anfn.

...

=(aaa ... to n factors) (bbb ..... to n factors)

In like manner, (abc)" = a"b"c"; and so on.

4. The converse of the principle of Art. 3 is evidently true. That is,

ambm = (ab)TM; ambmcm = (abc)"; etc.

5. The principles of Arts. 2-3 prove the method, already given in Ch. V., Art. 5, of raising a monomial to any required power.

Raise the numerical coefficient to the required power, and multiply the exponent of each literal factor by the exponent of the required power.

Ex. 1.

Ex. 2.

(4 a3b)2 = 42 a3×2b2 = 16 aob2.

(− 3 a1x2)3 = (− 3)3 a1×3 ̧2×3 —— 27 a12x6.

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small]

These examples illustrate the following method of raising any fraction to a required power:

Raise each term of the fraction to the required power; or, stated symbolically,

n

For,

b

n

an bn

[blocks in formation]
[merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

Write the squares, the cubes, and the nth powers of:

3 ab3

13. am+1. 14. xm-2. 15. 2x+y. 16. 3a-ly3.

Find the values of each of the following powers:

[ocr errors][merged small][merged small]
[ocr errors]
[ocr errors][merged small][merged small]
[blocks in formation]

25.

2 xyz)

Powers of Binomials.

7. By actual multiplication, we obtain

(a + b)3 = (a2 + 2 ab + b2) (a + b) = a3 + 3 a2b +3 ab2 + b3,

(a — b)3 = (a2 — 2 ab + b2) (a − b) = a3 − 3 a2b + 3 ab2 — b3,

(a + b)1 = (a2 + 2 ab + b2) (a2 +2ab+b2) = a1+4a3b+6a2b2+4ab3+b+,

(a - b)+= a+ - 4a3b+6a2b2 - 4 ab3 +b+.

The result of performing the indicated operation in a power of a binomial is called the Expansion of that power of the binomial.

In the preceding expansions the following laws are evident: (i.) The number of terms exceeds the binomial exponent by 1. (ii.) The exponent of a in the first term is equal to the binomial exponent, and decreases by 1 from term to term.

(iii.) The exponent of b in the second term is 1 and increases by 1 from term to term, and in the last term is equal to the binomial exponent.

(iv.) The coefficient of the first term is 1, and that of the second term, except for sign, is equal to the binomial exponent.

(v.) The coefficient of any term after the second is obtained, except for sign, by multiplying the coefficient of the preceding term by the exponent of a in that term, and dividing the product by a number greater by 1 than the exponent of b in that term.

E.g., the coefficient of the fourth term in the expansion of

[blocks in formation]

(vi.) The signs of the terms are all positive when the terms of the binomial are both positive; the signs of the terms alternate, + and -, when one of the terms of the binomial is negative.

Observe, as a check :

(vii.) The sum of the exponents of a and b in any term is equal to the binomial exponent.

(viii.) The coefficients of two terms equally distant from the beginning and the end of the expansion are equal.

In a subsequent chapter the above laws will be proved to hold for any positive integral power of the binomial.

8. Ex. 1.

(2 a − 3 b)1 = (2 a)* − 4 (2 a)3 (3 b) +6 (2 a)2 (3 b)2

=

[blocks in formation]

16a-96 ab+216 ab2-216 ab3+81 b1.

Ex. 2. (x+2y)5=x13+5 x1(2 y)+10 x3 (2 y)2

+10x2(2 y)3+5x(2 y)*+ (2 y)5

= x2+10x1y+40 x3y2+80 x2y3 +80 xy*+32 y3.

[blocks in formation]

Powers of Multinomials.

9. We have

(a+b+c)2 = [(a + b) + c]2 = (a + b)2 + 2 (a + b) c + c2

= a2 + 2 ab + b2 + 2 ac + 2 bc + c2.

(a+b+c)2 = a2 + b2 + c2 + 2 ab + 2 ac + 2 bc.

Therefore

In like manner,

(a

(a+b-c)2= a2 + b2 + c2 + 2 ab 2 ac- 2 bc.

[ocr errors]

· b − c)2 = a2 + b2 + c2 - 2 ab − 2 ac + 2 bc.

By repeated application of this principle we can obtain the square of a multinomial of any number of terms.

(a + b + c + d)2 = [(a + b + c)2 + d]2

We have

= a2 + b2 + c2+2ab+ 2 ac + 2 be + 2(a+b+c)d + d2 = a2 + b2 + c2 + d2 + 2 ab + 2 ac + 2 ad + 2 bc + 2 bd + 2 cd.

That is, the square of a multinomial is equal to the sum of the squares of the terms, plus the algebraic sum of twice the product of each term by each term which follows it.

Ex.1. (3x+5y—7 z)2=(3x)2+(5 y)2 + (−7 z)2+2(3x) (5 y)

+2(3x)(−7z)+2(5 y) (−7 z)

=9x2+25 y2+49 z2+30 xy−42 xz —70 yz.

EXERCISES III.

Raise each of the following expressions to the required.

[blocks in formation]

CHAPTER XIV.

EVOLUTION.

1. A Root of a number is one of the equal factors of the number.

E.g., 2 is a root of 4, of 8, of 16, etc.

2. A Second, or Square Root of a number is one of two equal factors of the number.

E.g., since 5 x 5 = 25 and (-5) (-5)= 25, therefore +5 and 5 are square roots of 25.

A Third, or Cube Root of a number is one of three equal factors of the number.

E.g., since 3 x 3 x 3 = 27, therefore 3 is since (-3) (-3) (-3)=-27, therefore

of 27.

[ocr errors]

a cube root of 27;

-

3 is a cube root

In general, the qth root of a number is one of q equal factors of the number.

E.g., a qth root of x is x.

3. The Radical Sign, √, is used to denote a root, and is placed before the number whose root is to be found.

The Radicand is the number whose root is required.

The Index of a root is the number which indicates what root is to be found, and is written over the radical sign. The index 2 is usually omitted.

E.g., 2/9, or √9, denotes a second, or square root of 9; the radicand is 9, and the index is 2.

4. A vinculum is often used in connection with the radical sign to indicate what part of an expression following the sign is affected by it.

« AnteriorContinuar »