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10. Find the sixth power of -2a2b3x1. 11. Find the seventh power of 2a2x3y. 12. Find the mth power of ab2x3.

186. Powers of Fractions.-Let it be required to find the

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From the rule for the multiplication of fractions, we have (2ab23 2ab2 2ab2 2ab2 8a3b6

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In a similar manner any fraction may be raised to any power. Hence, to raise a fraction to any power, we have the following

RULE.

Raise both numerator and denominator to the required power.

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187. Negative Exponents.-The rule of Art. 184, for raising a monomial to any power, holds true when the exponents of any of the letters are negative, and also when the exponent of the required power is negative.

Let it be required to find the square of a-3. This expres

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sion may be written which, raised to the second power, be

1

comes or a-6, the same result as would be obtained by

multiplying the exponent -3 by 2.

Also, let it be required to find that power of 2am2 whose exponent is -3.

The expression (2am2)-3 may be written.

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which

(2am2)3

Transferring the factors to the numerator, we

2-3a-3m-6, or a-3m-6.

EXAMPLES.

Find the value of each of the following expressions.

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188. Powers of Polynomials.—A polynomial may be raised to any power by the process of continued multiplication. If the quantity be multiplied by itself, the product will be the second power; if the second power be multiplied by the original quantity, the product will be the third power, and so on. Hence we have the following

RULE.

Multiply the quantity by itself until it has been taken as a factor as many times as there are units in the exponent of the required power.

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189. Square of a Polynomial.-We have seen, Art. 66, that the square of a binomial may be formed without the labor of actual multiplication. The same principle may be extended to polynomials of any number of terms. By actual multiplication, we find the square of a+b+c to be

a2+b2+c2+2ab+2ac+2bc;

that is, the square of a trinomial consists of the square of each term, together with twice the product of all the terms multiplied together two and two.

In the same manner we find the square of a+b+c+d to be

a2+b2+c2+d2+2ab+2ac+2ad+2bc+2bd+2cd;

that is, the square of any polynomial consists of the square of each term, together with twice the sum of the products of all the terms multiplied together two and two.

EXAMPLES.

1. Find the square of a+b+c+d+x.
2. Find the square of a-b+c.
3. Find the square of 1+2x+3x2.
4. Find the square of 1-x+x2-x3.
5. Find the square of a-2b+3ab—m.
6. Find the square of 1-3x+3x2-x3.
7. Find the square of a-2b+3c-4d.

In Chapter XVIII. will be given a method by which any power of a binomial may be obtained without the labor of multiplication.

CHAPTER XII.

EVOLUTION.

190. A root of a quantity is one of the equal factors which, multiplied together, will produce that quantity.

If a quantity be resolved into two equal factors, one of them is called the square root.

If a quantity be resolved into three equal factors, one of them is called the cube root.

If a quantity be resolved into four equal factors, one of them is called the fourth root, and so on.

191. Evolution is the process of extracting any root of a given quantity.

Evolution is indicated by the radical sign √.

Thus, a denotes the square root of a.

Va denotes the cube root of a.

Va denotes the nth root of a.

192. Surds.-When a root of an algebraic quantity which is required can not be exactly obtained, it is called an irrational or surd quantity.

Thus, Va is called a surd. √3 is also a surd, because the square root of 3 can not be expressed in numbers with perfect

exactness.

A rational quantity is one which can be expressed in finite terms, and without any radical sign; as, a, 5a2, etc.

193. An imaginary root is one which can not be extracted on account of the sign of the given quantity. Thus the square root of —4 is impossible, because no quantity raised to an even power can produce a negative result.

A root which is not imaginary is said to be real.

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