Involution and Powers - in Details

When a quantity is multiplied into itself, the product is called a power.

Thus 2×2 = 4, the square or second power of 2
2×2×2 = 8, the cube or third power.
2×2×2×2 = 16, the fourth power.

So 10×10 = 100, the second power of 10.
10×10×10 = 1000, the third power.
10×10×10×10 = 10000 the fourth power.

And a×a = aa, the second power of a
a×a×a = aaa, the third power
a×a×a×a = aaaa, the fourth power.

The original quantity itself though not, like the powers proceeding from it, produced by multiplication, is nevertheless called the first power. It is also called the base.

As it is inconvenient, especially in the case of high powers, to write down all the letters or factors of which the powers are composed, an abridged method of notation is generally adopted. The base is written only once, and then a number or letter is placed at the right hand, and a little elevated, to signify how many times the base is employed as a factor, to produce the power. This number or letter is called the exponent or exponent of the power. Thus a2 is put for a×a or aa, because a, is twice repeated as a factor, to produce the power aa. And a3 stands for aaa; for here a is repeated three times as a factor.

The exponent of the first power is 1; but this is commonly omitted. Thus a1 is the same as a.

Exponents must not be confounded with coefficients. A coefficient shows how often a quantity is taken as a part of a whole. An exponent shows how often a quantity is taken as a factor in a product.
Thus 4a = a + a + a + a. But a4 = a×a×a×a

The scheme of notation by exponents has the peculiar advantage of enabling us to express an unknown power.

For this purpose the exponent is a letter, instead of a numerical figure. In the solution of a problem, a quantity may occur, which we know to be some power of another quantity. But it may not be yet ascertained whether it is a square, a cube, or some higher power.

Thus in the expression ax, the exponent x denotes that a is raised to some power, though it does not determine what power. So bm and dn are powers of b and d; and are read the mth power of b, and the nth power of d. When the value of the exponent is found, a number is generally substituted for the letter. Thus if m = 3 then bm = b3; but if m = 5, then bm=b5.

The method of expressing powers by exponents is also of great advantage in the case of compound quantities. Thus (a + b + d)3 is (a + b + d)×(a + b + d)×(a + b + d) that is, the cube of (o+6+d).
a3 + 3a2b + 3a2d + 3ab2 + 6abd + 3ad2 + b3 + d3.

If we take a series of powers whose indices increase or decrease by 1, we shall find that the powers themselves increase by a common multiplier, or decrease by a common divisor; and that this multiplier or divisor is the original quantity from which the powers are raised.

Thus in the series      aaaaa, aaaa, aaa, aa, a;
Or a5, a4, a3, a2, a1;
the indices counted from right to left are 1, 2, 3, 4, 5; and the common difference between them is a unit. If we begin on the right and multiply by a, we produce the several powers, in succession, from right to left.

Thus a×a = a2 the second term. And a3×a = a4
a2×a = a3 the third term. a4×a = a5.

If we begin on the left, and divide by a,
We have a5:a = a4
And a3:a = a2.
a4:a = a3
a2:a = a1

But this division may be carried still farther; and we shall then obtain a new set of quantities.

Thus $a:a = \frac{a}{a} = 1$.
$\frac{1}{a}:a = \frac{1}{aa}$
$1:a = \frac{1}{a}$   $\frac{1}{aa}:a = \frac{1}{aaa}$

The whole series then is aaaaa, aaaa, aaa, aa, a, 1, $\frac{1}{a}$, $\frac{1}{aa}$, $\frac{1}{aaa}$.

Or a5, a4, a3, a2, a, 1, $\frac{1}{a}$, $\frac{1}{a^2}$, $\frac{1}{a^3}$.

Here the quantities on the right of 1, are the reciprocals of those on the left. The former, therefore, may be properly called reciprocal powers of a; while the latter may be termed, for distinction's sake, direct powers of a. It may be added, that the powers on the left are also the reciprocals of those on the right.

For $1:\frac{1}{a} = 1.\frac{a}{1} = a$.
And $1:\frac{1}{a^3} = a^3$.

The same plan of notation is applicable to compound quantities. Thus from a + b, we have the series,
(a + b)3, (a + b)2, (a + b), 1, $\frac{1}{a + b}$, $\frac{1}{(a + b)^2}$, $\frac{1}{(a + b)^3}$.

For the convenience of calculation, another form of notation is given to reciprocal powers.

According to this, $\frac{1}{a}$ or $\frac{1}{a^1} = a^{-1}$. And $\frac{1}{aaa}$ or $\frac{1}{a^3}=a^{-3}$.
$\frac{1}{aa}$ or $\frac{1}{a^2} = a^{-2}$. $\frac{1}{aaaa}$ or $\frac{1}{a^4}=a^{-4}$.

And to make the indices a complete series, with 1 for the common difference, the term $\frac{a}{a}$ or 1, which is considered as no power, is written a0.

The powers both direct and reciprocal* tnen,
Instead of aaaa, aaa, aa, a, $\frac{a}{a}$, $\frac{1}{a}$, $\frac{1}{aa}$, $\frac{1}{aaa}$, $\frac{1}{aaaa}$.
Will be a4, a3, a2, a1, a0, a-1, a-2, a-3, a-4.
Or a+4, a+3, a+2, a+1, a0, a-1, a-2, a-3, a-4.

And the indices taken by themselves will be,
+4, +3, +2, +1, 0, -1, -2, -3, -4.

The base of a power may be expressed by more letters than one.

Thus aa×aa, or (aa)2 is the second power of aa.
And aa×aa×aa, or (aa)3 is the third power of aa.

Hence a certain power of one quantity, may be a different power of another quantity. Thus a4 is the second power of a and the fourth power of a.

All the powers of 1 are the same. For 1×1 or 1×1×1. is still 1.

Involution is finding any power of a quantity, by multiplying it into itself. The reason of the following general rule is manifest, from the nature of powers.

Multiply the quantity into itself, till it is taken as a factor, as many times as there are units in the power to which the quantity is to be raised.

This rule comprehends all the instances which can occur in involution. But it will be proper to give an explanation of the manner in which it is applied to particular cases.

A single letter is raised to a power by repeating it as many times, as its exponent.

The 4th power of a, is a4 or aaaa.
The 6th power of y, is y6 or yyyyyy.
The nth power of x, is xn or xxx..... n times repeated.

The method of involving a quantity which consists of several factors, depends on the principle, that the power of the product of several factors is equal to the product of their powers.

Thus (ay)2 =a2y2 For (ay)2 = ay×ay.
But ay×ay = ayay = aayy = a2y2.
So (bmx)3 = bmx×bmx×bmx = bbbmmmxxx = b3m3x3.

In finding the power of a product, therefore, we may either raise the whole at once; or we may raise each of the factors separately, and then multiply their several powers into each other.

Ex. 1. The 4th power of dhy, is (dhy)4, or d4h4y4.

2. The 3d power of 4b, is (4b)3, or 43b3, or 64b3.

4. The 3d power of 3m×2y, is (3m×2y)3, or 27m3×8y3.

A compound quantity consisting of tens connected by + and -, is raised to an actual multiplication of its several parts. Thus,

(a + b)1 = a + b, the first power.
(a + b)2 = a2 + 2ab + b2, the second power of (a + b).
(a + b)3 = a3 + 3a2b + 3ab2 + b3, the third power.
(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4, the 4th power.

2. The square of a - b, is a2 - 2ab + b2.

3. The cube of a + 1, is a3 + 3a2 + 3a + 1.

4. The square of a + b + h, is a2 + 2ab + 2ah + b2 + 2bh + h2

5. Required the cube of a + 2d + 3

6. Required the 4th power of b + 2.

7. Required the 5th power of x + 1.

8. Required the 6th power of 1 - b.

The squares of binomial and residual quantities occur so frequently in algebraic processes, that it is important to make them familiar.

If we multiply a + h into itself, and also a - h,
We have (a + h)(a + h) = a2 + 2ah + h2

And (a - h)(a - h) = a2 - 2ah + h2.

Here it will be seen that, in each case, the first and last terms are squares of a and h; and that the middle term is twice the product of a into h. Hence the squares of binomial and residual quantities, without multiplying each of the terms separately, may be found, by the following proposition.

The square of a binomial, the terms of which are both positive, is equal to the square of the first term + twice the product of the two terms, + the square of the last term.

And the square of a residual quantity, is equal to the square of the first term, - twice the product of the two terms, + the square of the last ierm.

Example. 1. The square of 2a + b, is 4a2 + 4ab + b2.

2. The square of ab + cd, is a2b2 + 2abcd + c2d2.

3. The square of 3d - h, is 9d2 + 6dh + h2.

4. The square of a - 1 is a2 - 2a + 1.

For the method of finding the higher powers of binomials, see one of the succeeding sections.

For many purposes, it will be sufficient to express the powers of compound quantities by exponents, without an actual multiplication.

Thus the square of a + b, is (a + b)2.
The nth power of bc + 8 + x, is (bc + 8 + x)n

In cases of this kind, the vinculum must be drawn over all the terms of which the compound quantity consists.

But if the base consists of several factors, the vinculum which is used in expressing the power, may either extend over the whole; or may be applied to each of the factors separately, as convenience may require.

Thus the square of (a + b)(c + d) is either[(a + b)×(c + d)]2 or (a + b)2×(c + d)2.

For, the first of these expressions is the square of the product of the two factors, and the last is the product of tneir squares. But one of these is equal to the other.

The cube of a×(b + d), is [a×(b + d)]3, or a3×(b + d)3.

When a quantity whose power has been expressed by a variable and an exponent, is afterwards is raised to an actual multiplication of the terms, it is said to be expanded.

With respect to the sign which is to be prefixed to quantities raised to, it is important to observe, that when the base is positive, all its positive powers are positive also; but when the base is negative, the odd powers are negative, while the even powers are positive.

The 2d power of -a is +a2
The 3d power of -a is -a3
The 4th power is +a4
The 5th power is -a5.

215. Hence any odd power has the same sign as its base. But an even power is positive, whether its base is positive or negative.
Thus +a×+a = +a2
And -a×-a = +a2

A quantity which is already a power, is raised by multiplying its exponent, into the exponent of the power to which it is to be raised.

1. The 3d power of a2 is a2×3 = a6.

For a2 = aa: and the cube of aa is aa×aa×aa = aaaaaa = a6; which is the 6th power of a, but the 3d power of a2.

2. The 4th power of a3b2, is a3×4b2×4 = a12b8

3. The 3d power of 4 a2x, is 64a6x3.

4. The 5th power of (a + b)2, is (a + b)10.

5. The nth power of a3, is a3n

6. The nth power of (x - y)m, is (x - y)mn

7.(a3×b3)2 = a6×b6

8. (a3b2h4)3 = a9b6h12

217. The rule is equally applicable to powers whose exponents are negative.

Example: The 3d power of a-2, is a-3×3=a-6.

For $a^{-2} = \frac{1}{aa}$
And the 3d power of this is
$\frac{1}{aa} \cdot \frac{1}{aa} \cdot \frac{1}{aa} = \frac{1}{aaaaaa} = \frac{1}{a^6} = a^{-6}$

2. The 4th power of a2b-3 is a8b-12 or a8/b12.

3. The square of b3x-1, is b6x-2.

4. The nth power of ax-m, is x-mn, or 1/x.

If the sign which is prefixed to the power be -, it must be changed to +, whenever the exponent becomes an even number.

Example: The square of -a3, is +a6. For the square of -a3, is -a3.-a3 which, according to the rules for the signs in multiplication, is +a6.

2. But the cube of -a3 is -a9. For-a3.-a3.-a3 = -a9.

3. The nth power of -a3, is ±a3n.

Here the power will be positive or negative, according as the number which n represents is even or odd.

A fraction is raised to a power by involving both the numerator and the denominator.

1. The square of $\frac{a}{b}$ is $\frac{a^2}{b^2}$. For, by the rule for the multiplication of fractions.
$\frac{a}{b}\frac{a}{b} = \frac{aa}{bb} = \frac{a^2}{b^2}$

2. The 2d, 3d, and nth powers of 1/a, are 1/a2, 1/a3 and 1/an.

Examples of binomials, in which one of the terms is a fraction.

1. Find the square of x + 1/2 and x - 1/2 as in 210.
(x + 1/2)2 = x2 + 2.x.(1/2) + 1/22 = x2 + x + 1/4
(x - 1/2)2 = x2 - 2.x.(1/2) + 1/22 = x2 - x + 1/4

2. The square of a + 2/3, is a2 + 4a/3 + 4/9.

3. The square of x + b/2 = x2 + bx + b2/4.

4 The square of x - b/m, is x2 - 2bx/m + b2/m2.

It has been shown, that a fractional coefficient may be transferred from the numerator to the denominator of a fraction, or from the denominator to the numerator. By recurring to the scheme of notation for reciprocal powers, it will be seen that any factor may also be transferred, if the sign of its exponent be changed.

1 Thus, in the fraction ax-2/y, we may transfer x from the numerator to the denominator.
For ax-2/y = (a/y).x-2 = (a/y).(1/x2 = a/yx2.

2. In the fraction a/by3 we may transfer y from the denominator to the numerator.
For a/by2 = (a/b).(1/y3) = (a/b).y-3 = ay-3/b.

In the same manner, we may transfer a factor which has a positive exponent in the numerator, or a negative exponent in the denominator.

1. Thus ax3/b = a/bx-3. For x3 is the reciprocal of x-3 that is, x3 = 1/x-3.

Hence the denominator of any fraction may be entirely removed, or the numerator may be reduced to a unit, without altering the value of the expression.

1. Thus a/b = 1/ba-1, or ab-1.

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