Addition, Subtraction, Multiplication and Divison of Powers

Addition and Subtraction of Powers

It is obvious that powers may be added, like other quantities, by uniting them one after another with their signs.

Thus the sum of a3 and b2, is a3 + b.
And the sum of a3 - bn and h5 -d4 is a3 - bn + h5 - d4.

The same powers of the same letters are like quantities and their coefficients maybe added or subtracted.

Thus the sum of 2a2 and 3a2, is 5a2.

It is as evident that twice the square of a, and three times the square of a, are five times the square of a, as that twice a and three times a, are five times a.

But powers of different letters and different powers of the same letter, must be added by writing them down with their signs.

The sum of a2 and a3 is a2 + a3.

It is evident that the square of a, and the cube of a, are neither twice the square of a, nor twice the cube of a.

The sum of a3bn and 3a5b6 is a3bn + 3a5b6.

Subtraction of powers is to be performed in the same manner as addition, except that the signs of the subtrahend are to be changed.

From 2a4 3h2b6 5(a - h)6
Sub. -6a4 4h2b6 2(a - h)6
Diff. 8a4 -h2b6 3(a - h)6

OR: 2a4 - (-6a4) = 8a4
3h2b6 - 4h2b6 = -h2b6
5(a - h)6 - 2(a - h)6 = 3(a - h)6

Multiplication of Powers

Powers may be multiplied, like other quantities, by writing the factors one after another, either with, or without, the sign of multiplication between them.

Thus the product of a3 into b2 is a3b2 or aaabb.

Mult. x-3 3a6y2 a2b3y2
Into am -2x a3b2y
Prod. amx-3 -6a6xy2 a2b3y2a3b2y

x-3 × am = amx-3
3a6y2 × (-2x) = -6a6xy2
a2b3y2 × a3b2y = a2b3y2a3b2y

The product in the last example, may be abridged, by bringing together the letters which are repeated.
It will then become a5b5y3.

The reason of this will be evident, by recurring to the series of powers in Art. 204, viz.

By comparing the several terms with each other, it will be seen that if any two or more of them be multiplied together, their product will be a power whose exponent is the sum of the exponents of the factors.

Thus a2⋅a3 = aa⋅aaa = aaaaa = a5.

Here 5, the exponent of the product, is equal to 2 + 3, the sum of the exponents of the factors.

So an⋅am = am+n.

For an, is a taken for a factor as many times as there are units in n;

And am, is a taken for a factor as many times as there are units in m;

Therefore the product must be a taken for a factor as many times as there are units in both m and n. Hence,

Powers of the same root may be multiplied, by adding their exponents.

This a2⋅a6 = a2+6 = a8. And x3⋅x2⋅x = x3+2+1 = x6.

Multiply 4an b2y3 (b + h - y)n
Into 2an b4y (b + h - y)
Product 8a2n b6y4 (b + h - y)n+1

4an × 2an = 8a2n
b2y3 × b4y = b6y4
(b + h - y)n × (b + h - y) = (b + h - y)n+1

Mult. x3 + x2y + xy2 + y3 into x - y.
Answer: x4 - y4.
Mult. x3 + x - 5 into 2x3 + x + 1.

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

1. Thus a-2.a-3 = a-5. That is $\frac{1}{aa}\cdot \frac{1}{aaa}=\frac{1}{aaaaa}$.

2. y-n⋅y-m = y-n-m.

3. a-n⋅am = am-n.

If a + b be multiplied into a - b, the product will be a2 - b2: that is

The product of the sum and difference of two quantities, is equal to the difference of their squares.

If the sum and difference of the squares be multiplied, the product will be equal to the difference of the fourth powers.

Thus (a - y)⋅(a + y) = a2 - y2.
(a2 - y2)⋅(a2 + y2) = a4 - y4.
(a4 - y4)⋅(a4 + y4) = a8 - y8.

Division of Powers

Powers may be divided, like other quantities, by rejecting from the dividend a factor equal to the divisor; or by placing the divisor under the dividend, in the form of a fraction.

Thus the quotient of a3b2 divided by b2, is a3.

Divide 9a3y4 a2by + 3a2y2 d⋅(a - h + y)3
By -3a3 a2y (a - h + y)3
Quotient -3y4 b + 3y d

The quotient of a5 divided by a3, is a5/a3. But this is equal a2. For, in the series
     a+4, a+3, a+2, a+1, a0, a-1, a-2, a-3, a-4.
if any term be divided by another, the index of the quotient will be equal to the difference between the index of the dividend and that of the divisor.

A power may be divided by another power of the same root, by subtracting the index of the divisor from that of the dividend.

Thus y3:y2 = y3-2 = y1. That is yyy/yy = y.

And an+1:a = an+1-1 = an. That is aan/a = an.

Divide y2m 8an+m 12(b + y)n
By ym 4am 3(b + y)3
Quot. ym 2an 4(b +y)n-3

The rule is equally applicable to powers whose exponents are negative.
The quotient of a-5 by a-3, is a-2.
That is $\frac{1}{aaaaa}:\frac{1}{aaa} = \frac{1}{aaaaa} \cdot \frac{aaa}{1} = \frac{aaa}{aaaaa} = \frac{1}{aa}$.

3. h2:h-1 = h2+1 = h3. That is h2:(1/h) = h2⋅(h/1) = h3

The multiplication and division of powers, by adding and subtracting their indices, should be made very familiar; as they have numerous and important applications, in the higher branches of algebra.

Examples of Fractions Containing Powers

In the section on fractions, the following examples were omitted for the sake of avoiding an anticipation of the subject of powers.

1. Reduce 5a4/3a2 to lower terms. Answer: 5a2/3.

2. Reduce 6x6/3x5 to lower terms. Answer: 2x/1 or 2x.

3. Reduce a2/a3 and a-3/a-4 to a common denominator.
a2⋅a-4 is a-2 the first numerator. (Art. 146.)
a3⋅a-3 is a0 = 1, the second numerator.
a3⋅a-4 is a-1, the common denominator.
The fractions reduced are therefore a-2/a-1 and 1/a-1.

4. Reduce 2a4/5a3 and a2/a4, to a common denominator.
Answer: 2a3/5a7 and 5a5/5a7 or 2a3/5a2 and 5/5a2. (Art. 142.)

5. Multiply (a3 + b)/b4, into (a - b)/3.

6. Multiply (a5 + 1)/x2, into (b2 - 1)/(x + a).

7. Multiply b4/a-2 into h-3/x, and an/y-3.

8. Divide a4/y3, by a3/y2. Answer: a/y.

9. Divide (h3 - 1)/d4, by (dn + 1)/h.

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