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EVOLUTION.

208. We know that there are two square roots of a2, namely a; we also know [Art. 188] that there are three cube roots of a3, of which a is one, and the other two are imaginary.

There is therefore an important difference between powers and roots; for there is only one nth power, but there is more than one nth root, of a given expression.

209. An expression which when raised to the nth power, where n is any positive integer, becomes equal to a given expression, is called an nth root of the given expression.

We have shown in Art. 199 that the mth power of a product is the product of the mth powers of its factors; hence, conversely, the mth root of a product is the product of the mth roots of its factors.*

Thus

and

√abe = √a √b vc,

Wab = w/a w/b.

Again, we have shown in Art. 200 that the nth power of a monomial expression is obtained by multiplying the index of each of its factors by n.

It follows conversely that if we divide the index of each factor of a given expression by n, we shall obtain an nth root of the expression. For by raising to the nth

* It should be noticed that the proof, in Art. 52, that the factors of a product may be taken in any order, only holds good when those factors represent integral or fractional numbers, and does not enable us to assert that va x √b = √b × √a, when va or vb is really a surd. For a proof that surds obey the Fundamental Laws of Algebra see Treatise on Algebra, Art. 162.

power the result obtained by such division of the indices, we must clearly get the original expression.

Thus one value of /at is a2, one value of Vab12c9 is a2b1c3, and one value of Wanpbng is arbi.

When the square root of an expression which is not a perfect square, or the cube root of an expression which. is not a perfect cube, is required, the operation cannot be performed. We can, for example, only write the square root of a as √a, and the cube root of a2 as a2, and similarly in other cases.

SQUARE ROOT.

210. We now proceed to consider the square root of multimonial expressions.

In Art. 114 we have shown how to write down the square root of any trinomial expression which is a complete square.

Having arranged the expression according to ascending or descending powers of some letter, the square root of the whole expression is then found by taking the square roots of the extreme terms with the same or with different signs according as the sign of the middle term is positive or negative.

Thus, to find the square root of

4a8 - 12 a1b3 +966.

The square roots of the extreme terms are

2a and ± 363.

Hence, the middle term being negative, the required (2a-3b3).

square root is

NOTE. In future only one of the square roots of an expression will be given, namely that one for which the sign of the first term is positive to find the other root all the signs must be changed.

and

:

As other examples

√(49 a10 + 28 a3 + 4) = 7 a3 + 2

√(1+5xy3 + 25 x2y6) = 1 + 1⁄2 xy3,

√{a2 + 2 a(b + c) + (b + c)2} = a + b + c.

211. When an expression which contains only two different powers of a particular letter is arranged according to ascending or descending powers of the letter, it will only contain three terms. For example, the expression

a2+b2+c2+ 2 bc + 2 ca + 2 ab,

which contains no other power of a but a2 and a, when arranged according to powers of a, is

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

Thus any expression which only contains two different powers of a particular letter can be written as a trinomial expression; and since we can write down the square root of any trimonial expression which is a complete square, it follows that the square root of any expression which is a complete square can be written down by inspection, provided that the expression only contains two different powers of some particular letter.

For example, to find the square root of

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

Arranging the expression according to the descending powers of a, we have a2 + 2 a(b+c) + (b2 + 2 bc + c2),

that is, which is

a2+2a(b+c) + (b + c)2,

{a + (b + c)}2.

Thus

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

Also, to find the square root of

x2 + 4 y1 + 9 z1 + 4x2y2 — 6 x2z2 – 12 y2z2.

Arrange the expression according to descending powers of x; we then have

that is,

which is

Thus

x2 + 2x2(2 y2 - 3 z2)+4y2+9 zł - 12 y2z2,
x+2x2(2 y2-322)+(2 y2-322)2,
{x2 + (2 y2 − 3 z2)}2.

√(x2 + 4 y1 + 9 z1 + 4 x2y2 — 6 x2z2 — 12 y2z2)= (x2 + 2 y2 − 3 z2). Again, to find the square root of

a2 + 2 abx+(b2 + 2 ac) x2 + 2 bcx3 + c2x2.

Arrange the expression according to powers of a; we then have a2 + 2 a(bx + cx2) + b2x2 + 2 bcx3 + c2x2,

that is, which is

a2 + 2 a(bx + cx2)+ (bx + cx2)2,

{a + (bx + cx2)}2.

Hence the required square root is (a + bx + cx2).

And to find the square root of

x6 - 2x5 + 3x1 + 2 x3 (y − 1) + x2(1 − 2 y) + 2xy + y2. The expression only contains y2 and y; we therefore arrange it according to powers of y; we then have

y2+2 y(x3- x2 + x) + x6 − 2 x5 + 3 x4 -- 2 x3 + x2.

Now if the expression is a complete square at all, the last of the three terms must be the square of half the coefficient of y; and it is easy to verify that

x6 - 2x5 + 3x4 - 2 x3 + x2 is (x3 — x2 + x)2.

Thus the given expression is

y2 + 2 y(x3 — x2 + x) + (x3 — x2 + x)2.

The required square root is therefore

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From the above it will be seen that however many terms there may be in an expression which is a perfect square, the square root can be written down by inspection, provided only that the expression contains only two dif ferent powers of some particular letter.

EXAMPLES LXIII.

Write down the square roots of the following expressions.

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16. 25 at +9 ba + ca . 12 b2c2+ 20 c2a2 - 30 a2b2.

212. In order to show how to find the square root of any algebraical expression, we will take an expression and form its square, and then show how to reverse the process.

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