1.6 Homework: algebraic generalizations

In class, we found that if you multiply $2^6$ by 2, you get $2^7$ . If you multiply $2^{10}$ by 2, you get $2^{11}$ . We expressed this as a general rule that $\left(2^x\right)\left(2\right)=2^{x+1}$ .

Now, we’re going to make that rule even more general. Suppose I want to multiply $2^5$ times $2^3$ . Well, $2^5$ means $2\ast 2\ast 2\ast 2\ast 2$ , and $2^3$ means $$ $2\ast 2\ast 2$ . So we can write the whole thing out like this.

$(2^5)(2^3)=2^8$

• a Using a similar drawing, demonstrate what $({\text{10}}^3)({\text{10}}^4)$ must be.
• b Now, write an algebraic generalization for this rule.________________

The following statements are true.

• $3\times 4=4\times 3$
• $7\times -3=-3\times 7$
• $1/2\times 8=8\times 1/2$

Write an algebraic generalization for this rule.________________

In class, we talked about the following four pairs of statements.

• $8\times 8=64$
• $7\times 9=63$

• $5\times 5=25$
• $4\times 6=24$

• $10\times 10=100$
• $9\times 11=99$

• $3\times 3=9$
• $2\times 4=8$

• a You made an algebraic generalization about these statements: write that generalization again below. Now, we are going to generalize it further. Let’s focus on the $10\times 10$ thing. $10\times 10=100$ There are two numbers that are one away from 10; these numbers are, of course, 9 and 11. As we saw, $9\times 11$ is 99. It is one less than 100.
• b Now, suppose we look at the two numbers that are two away from 10? Or three away ? Or four away ? We get a sequence like this (fill in all the missing numbers):

  $10\times 10=100$
  $9\times 11=99$	1 away from 10, the product is 1 less than 100
  $8\times 12=\_\_\_\_$	2 away from 10, the product is ____ less than 100
  $7\times 13=\_\_\_\_$	3 away from 10, the product is ____ less than 100
  __×__=___	__ away from 10, the product is ____ less than 100
  __×__=___	__ away from 10, the product is ____ less than 100

• c Do you see the pattern? What would you expect to be the next sentence in this sequence?
• d Write the algebraic generalization for this rule.
• e Does that generalization work when the “___away from 10” is 0? Is a fraction? Is a negative number? Test all three cases. (Show your work!)