# Addition, Subtraction, Multiplication and Division 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 a^{3} and b^{2}, is a^{3} + b

And the sum of a^{3} - b^{n} and h^{5} -d^{4} is a^{3} - b^{n} + h^{5} - d^{4}.

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

^{2}and 3a

^{2}, is 5a

^{2}.

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 a^{2} and a^{3} is a^{2} + a^{3}.

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 a^{3}b^{n} and 3a^{5}b^{6} is a^{3}b^{n} + 3a^{5}b^{6}.

*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 | 2a^{4} |
3h^{2}b^{6} |
5(a - h)^{6} |

Sub. | -6a^{4} |
4h^{2}b^{6} |
2(a - h)^{6} |

Diff. | 8a^{4} |
-h^{2}b^{6} |
3(a - h)^{6} |

**OR:**
2a^{4} - (-6a^{4}) = 8a^{4}

3h^{2}b^{6} - 4h^{2}b^{6} = -h^{2}b^{6}

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 a^{3} into b^{2} is a^{3}b^{2} or aaabb.

Mult. | x^{-3} |
3a^{6}y^{2} |
a^{2}b^{3}y^{2} |

Into | a^{m} |
-2x | a^{3}b^{2}y |

Prod. | a^{m}x^{-3} |
-6a^{6}xy^{2} |
a^{2}b^{3}y^{2}a^{3}b^{2}y |

**OR:**

x^{-3} × a^{m} = a^{m}x^{-3}

3a^{6}y^{2} × (-2x) = -6a^{6}xy^{2}

a^{2}b^{3}y^{2} × a^{3}b^{2}y = a^{2}b^{3}y^{2}a^{3}b^{2}y

The product in the last example, may be abridged, by bringing together the letters which are repeated.

It will then become a^{5}b^{5}y^{3}.

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 a^{2}⋅a^{3} = aa⋅aaa = aaaaa = a^{5}.

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

So a^{n}⋅a^{m} = a^{m+n}.

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

And a^{m}, 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 base may be multiplied, by adding their exponents.**

This a^{2}⋅a^{6} = a^{2+6} = a^{8}. And x^{3}⋅x^{2}⋅x = x^{3+2+1} = x^{6}.

Multiply | 4a^{n} |
b^{2}y^{3} |
(b + h - y)^{n} |

Into | 2a^{n} |
b^{4}y |
(b + h - y) |

Product | 8a^{2n} |
b^{6}y^{4} |
(b + h - y)^{n+1} |

**OR:**

4a^{n} × 2a^{n} = 8a^{2n}

b^{2}y^{3} × b^{4}y = b^{6}y^{4}

(b + h - y)^{n} × (b + h - y) = (b + h - y)^{n+1}

Mult. x^{3} + x^{2}y + xy^{2} + y^{3} into x - y.

Answer: x^{4} - y^{4}.

Mult. x^{3} + x - 5 into 2x^{3} + 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}⋅a^{m} = a^{m-n}.

If a + b be multiplied into a - b, the product will be a^{2} - b^{2}: 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) = a^{2} - y^{2}.

(a^{2} - y^{2})⋅(a^{2} + y^{2}) = a^{4} - y^{4}.

(a^{4} - y^{4})⋅(a^{4} + y^{4}) = a^{8} - y^{8}.

### 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 a^{3}b^{2} divided by b^{2}, is a^{3}.

Divide | 9a^{3}y^{4} |
a^{2}by + 3a^{2}y^{2} |
d⋅(a - h + y)^{3} |

By | -3a^{3} |
a^{2}y |
(a - h + y)^{3} |

Quotient | -3y^{4} |
b + 3y | d |

The quotient of a^{5} divided by a^{3}, is a^{5}/a^{3}. But this is equal a^{2}. For, in the series

a^{+4}, a^{+3}, a^{+2}, a^{+1}, a^{0}, a^{-1}, a^{-2}, a^{-3}, a^{-4}.

if any term be divided by another, the exponent of the quotient will be equal to the *difference* between the exponent of the dividend and that of the divisor.

**A power may be divided by another power of the same base, by subtracting the exponent of the divisor from that of the dividend**.

Thus y^{3}:y^{2} = y^{3-2} = y^{1}. That is yyy/yy = y.

And a^{n+1}:a = a^{n+1-1} = a^{n}. That is aa^{n}/a = a^{n}.

Divide | y^{2m} |
8a^{n+m} |
12(b + y)^{n} |

By | y^{m} |
4a^{m} |
3(b + y)^{3} |

Quot. | y^{m} |
2a^{n} |
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. h^{2}:h^{-1} = h^{2+1} = h^{3}. That is h^{2}:(1/h) = h^{2}⋅(h/1) = h^{3}

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 5a^{4}/3a^{2} to lower terms. Answer: 5a^{2}/3.

2. Reduce 6x^{6}/3x^{5} to lower terms. Answer: 2x/1 or 2x.

3. Reduce a^{2}/a^{3} and a^{-3}/a^{-4} to a common denominator.

a^{2}⋅a^{-4} is a^{-2} the first numerator. (Art. 146.)

a^{3}⋅a^{-3} is a^{0} = 1, the second numerator.

a^{3}⋅a^{-4} is a^{-1}, the common denominator.

The fractions reduced are therefore a^{-2}/a^{-1} and 1/a^{-1}.

4. Reduce 2a^{4}/5a^{3} and a^{2}/a^{4}, to a common denominator.

Answer: 2a^{3}/5a^{7} and 5a^{5}/5a^{7} or 2a^{3}/5a^{2} and 5/5a^{2}. (Art. 142.)

5. Multiply (a^{3} + b)/b^{4}, into (a - b)/3.

6. Multiply (a^{5} + 1)/x^{2}, into (b^{2} - 1)/(x + a).

7. Multiply b^{4}/a^{-2} into h^{-3}/x, and a^{n}/y^{-3}.

8. Divide a^{4}/y^{3}, by a^{3}/y^{2}. Answer: a/y.

9. Divide (h^{3} - 1)/d^{4}, by (d^{n} + 1)/h.