Rules of exponents

Here are the first six powers of two.

• $2^1=2$
• $2^2=4$
• $2^3=8$
• $2^4=16$
• $2^5=32$
• $2^6=64$

• A

  If I asked you for $2^7$ (without a calculator), how would you get it? More generally, how do you always get from one term in this list to the next term ?

• B

  IWrite an algebraic generalization to represent this rule.

Suppose I want to multiply $2^5$ times $2^3$ . Well, $2^5$ means 2×2×2×2×2, and $2^3$ means 2×2×2. So we can write the whole thing out like this.

• A

  This shows that ( $2^{5}2^3=2^{\mathrm{[\; ]}}$

• B

  Using a similar drawing, demonstrate what $10^{3}10^4$ must be.

• C

  Now, write an algebraic generalization for this rule.

• D

  Show how your answer to 1b (the “getting from one power of two, to the next in line”) is a special case of the more general rule you came up with in 2c (“multiplying two exponents”).

Now we turn our attention to division. What is $\frac{3^{\text{12}}}{3^{\text{10}}}$ ?

• A

  Write it out explicitly. (Like earlier I wrote out explicitly what $2^{5}2^3$ was: expand the exponents into a big long fraction.)

• B

  Now, cancel all the like terms on the top and the bottom. (That is, divide the top and bottom by all the 3s they have in common.)

• C

  What you are left with is the answer. So fill this in: $\frac{3^{\text{12}}}{3^{\text{10}}}$ $3^{\mathrm{[\; ]}}$ .

• D

  Write a generalization that represents this rule.

• E

  Suppose we turn it upside-down. Now, we end up with some 3s on the bottom. Write it out explicitly and cancel 3s, as you did before: $\frac{3^{\text{10}}}{3^{\text{12}}}$ = ___________________________ = $\frac{1}{3^{\left[\right]}}$

• F

  Write a generalization for the rule in part (e). Be sure to mention when that generalization applies, as opposed to the one in part (d)!

Use all those generalizations to simplify $\frac{x^3{y}^3{x}^7}{x^5{y}^5}$

Now we’re going to raise exponents, to exponents. What is $2^3^4$ ? Well, $2^3$ means 2×2×2. And when you raise anything to the fourth power, you multiply it by itself, four times. So we’ll multiply that by itself four times:

$2^3^4$ = (2×2×2) (2×2×2) (2×2×2) (2×2×2)

• A

  So, just counting 2s, $2^3^4=2^{\mathrm{[\; ]}}$ .

• B

  Expand out $10^5^3$ in a similar way, and show what power of 10 it equals.

• C

  Find the algebraic generalization that represents this rule.