Exponents Doron ZEILBERGER^{1}

Algebra In algebra we have symbols and numbers mixed up. The notation is confusing. ab means a times b, i.e. a × b, but we don’t write the ×. But for numbers, we have to write the ×, since 23 means “twenty-three” and not 2 times 3. In algebra we have parantheses, which means do it first. They are very important. Example: 3(a + b) , means: first add up a and b then multiply by 3. They are rules for getting rid of parentheses, called foiling. In this case we have 3(a) + 3(b) = 3a + 3b. When we pronounce 3(a + b) we say “3 times the quantity a + b. Precedence: Mathematicians are lazy and often they don’t write parentheses explicity, but use conventions of precedence. First: All powers Second: All multiplications and divisions Third: all additions and subtractions So a + bc + d is really shorthand for a + (bc) + d. This is not to be confused with (a + b)(c + d) which means something else completely. Problem: Do (3 + 11)(8 − 4) + 7 · 2 − 8/2. Exponents: a^{n} means “a times a times a … times a”, where there are n of the a’s. You can always spell-out numerical or symbolic powers. Examples: 2^{3} = 2 × 2 × 2 = 8. a^{4} = aaaa. Important convention:

a^{−n} = \frac{1}{a^{n}}

1 Department of Mathematics, Rutgers University (New Brunswick), Hill Center-Busch Campus, 110 Frelinghuysen

Rd., Piscataway, NJ 08854-8019, USA. zeilberg at math dot rutgers dot edu , http://www.math.rutgers.edu/~zeilberg .

Examples:

2^{−3} = \frac{1}{2^{3}} = \frac{1}{8}

3^{−2} = \frac{1}{3^{2}} = \frac{1}{9}

(1/4)^{−2} = \frac{1}{(1/4)^{2}} = \frac{1}{1/16} = 16

Reminder: “The bottom of the bottom goes to the top”

\frac{1}{\frac{1}{W hatever}} = W hatever .

The Spelling-Out Method

\frac{3^{−2}}{2^{−2}} = \frac{1/9}{1/4} = 4/9 .

Watch Out from a Common Mistake: The following is wrong!!!!

\frac{3^{−2}}{2^{−2}} = \frac{3}{2} .

You can’t “cancel-out” the the same thing (in our case −2) when they are exponents. To be safe, always do the numerical part of exponent expressions by the spelling-out method. Examples: 5^{2} = 5 · 5 = 25 , (−5)^{2} = (−5) · (−5) = 25 (remember: minus times minus is plus), −5^{2} = −(5^{2}) = −25. Watch out: −5^{2} is not (−5)^{2}. By the precedence rules you first do the power, 5^{2} = 25 and only then you do the −, getting −25. −5^{2} is the same as −(5^{2}). On the other-hand, if they want you to do the − first and then do the power, then they need parantheses (−5)^{2}. Do right now: 2^{3} = (−2)^{4} = −2^{4} = −(−5)^{2} = −0.2^{3} = (−0.2)^{3} = (−\frac{3}{5})^{2} = Properties of Exponents Rule 1:

(ab)^{n} = a^{n}b^{n} .

Why is it true?: For example, when n = 3, we have (ab)^{3} = (ab)(ab)(ab) = ababab = aaabbb = a^{3}b^{3} .

Rule 2:

x^{n}x^{m} = x^{n+m} .

Why is it true?: For example, when n = 3, m = 4, we have x^{3}x^{4} = (xxx)(xxxx) = xxxxxxx = x^{7} .

Rule 3:

\frac{x^{n}}{x^{m}} = x^{n−m} .

Why is it true?: For example, when n = 5, m = 3, we have

\frac{x^{5}}{x^{3}} = \frac{xxxxx}{xxx} = xx = x^{2} .

Rule 4

(x^{m})^{n} = x^{mn} .

Why is it true?: For example, when n = 3, m = 4, we have (x^{3})^{4} = (x^{3})(x^{3})(x^{3})(x^{3}) = (xxx)(xxx)(xxx)(xxx) = xxxxxxxxxxxx = x^{12} .

Examples: x^{5} · x^{4} = x^{5+4} = x^{9} z^{2} · z^{5} = z^{2+5} = z^{7} (3^{2})^{5} = 3^{2·5} = 3^{10}

(−\frac{2}{3}x^{2})^{3} = (−1)^{3} \frac{2^{3}}{3^{3}} (x^{2})^{3} = −\frac{8}{27}x^{6} .

5a^{7}(−4a^{6}) = (5)(−4)a^{7} · a^{6} = −20a^{7+6} = −20a^{13} .

Common Mistake (Watch Out!):

(a^{6})(a^{7}) = a^{42}(W RONG!!!)

(a^{6})^{7} = a^{42} is right!, but if 6 and 7 are at the same level then (a^{6})(a^{7}) = a^{6+7} = a^{13}, i.e. you add the exponents not multiply them. But in

(a^{6})^{7}

the 7 is “above” the 6, so you multiply . Do right now! −3a^{2}(4a^{4}) = −3b^{3}a^{2}(3a^{5}b)(2ab^{2}) = (3^{3})^{4} = (2^{5})^{2} · (2^{2})^{5} · 2^{4} = Writing with positive exponents Remember

x^{−n} = \frac{1}{x^{n}} .

Examples:

3^{−2} = \frac{1}{3^{2}} = \frac{1}{9}

(−5)^{−2} = \frac{1}{(−5)^{2}} = \frac{1}{25}

(−2)^{−5} = \frac{1}{(−2)^{5}} = \frac{1}{−32} = −\frac{1}{32} .

(\frac{3}{4})^{−2} = \frac{3^{−2}}{4^{−2}} = \frac{4^{2}}{3^{2}} = \frac{16}{9} .

Do Right Now!:

(\frac{2}{3})^{−3} =

(\frac{1}{3})^{−2} + (\frac{1}{2})^{−3} =

(\frac{1}{4})^{−3} + (\frac{1}{4})^{−2} =

Simplifying complicated expressions Example

(5y^{4})^{−3}(2y^{−2})^{3} = (5^{−3}2^{3})y^{−12} · y^{−6} = \frac{8}{125}y^{−18} = \frac{8}{125y^{18}} .

Do right now!

(3z^{−2})^{3}(4z^{−3})^{2} =
(5w^{−4})^{2}(4w^{−5})^{3} =

Dividing Exponents Remember:

\frac{x^{m}}{x^{n}} = x^{m−n}

Careful: Don’t do things in your head! Do it step-by-step. It is very easy to make mistakes by doing “mental math”, remember “minus minus is plus”. Examples:

\frac{x^{−1}}{x^{9}} = x^{−1−9} = x^{−10} = \frac{1}{x^{10}} .

\frac{t^{−10}}{t^{−4}} = t^{−10−−4} = t^{−6} = \frac{1}{t^{6}} .

(\frac{8x^{2}y}{4x^{4}y^{−3}} )^{4} = (\frac{2x^{2}y}{x^{4}y^{−3}} )^{4} =

2^{4}(x^{−2}y^{4})^{4} = 16(x^{−8}y^{16}) = \frac{16y^{16}}{x^{8}} .

Do Right Now!

\frac{z^{−10}}{z^{−5}} =

\frac{w^{−3}}{w^{8}} =

\frac{x^{13}}{x^{18}} =

(\frac{6x^{3}y^{2}}{3xy^{−2}} )^{2} =

(\frac{16x^{−3}y^{−2}z^{3}}{4x^{−1}y^{2}z^{2}} )^{4} =

(\frac{9x^{3}y^{3}z^{3}}{3x^{−1}y^{−2}z^{−2}} )^{−1} =