Name ________________________________________________________ Date _______________ Hour ______

Standard G: Exponents

Lesson 2

Product & Quotient To A Power Rules

Warm-Up:

1. If you have a product of exponential expressions with the same base, what operation should you

do with the exponents?

2. If you have a quotient of exponential expressions with the same base, what operation should you

do with the exponents?

Simplify the following exponential expressions:

3. (2𝑥2𝑦3)(2𝑥2𝑦3) 4. 3𝑎2𝑏3𝑐10

12𝑎2𝑏5𝑐

5. (2𝑥2𝑦3)(22𝑥𝑦4)

42𝑥5𝑦9

6. Using your calculator, type the following expressions in exactly how you see them (including the

parentheses) and find their values. What do you think makes them have different values? (If you get

the same value, your answers are wrong).

a) (−2)4 b) −24

7. Using your calculator, type the following expressions in exactly how you see them (including the

parentheses) and find their values. Reflect back on #6 and see if you can figure out why the same

pattern doesn’t apply.

a) (−2)3 b) −23

Product to a Power: If you have a product of multiple exponential expressions all being raised to a

power, usually using parentheses, you should ____________________ all of the inside exponents by the

outside exponent.

• It is usually helpful for students to write in any missing exponents so that they don’t skip any of

the base terms.

• It is also extremely important to notice the name of the rule- this rule only works when you have

a product being raised to a power. In the next unit, we will discuss what happens when you

have a sum or difference being raised to a power.

• Remember that, using exponent rules, only the ___________________ will change- the __________

will not change. Additionally, you must expand out any number bases at the end of the

problem (something like 23 will turn into 8 at the end).

• Remember the order of operations- exponents come before multiplication, so if you have a

product to a power and an additional product/quotient in the problem, handle the product

to a power first.

Why? Remember that an exponent tells us how many times to multiply a base to itself. Using that

logic, rewrite and simplify the following exponential expression:

(3𝑥3𝑦4)2 =

Examples:

1. (𝑥2𝑦2)3 = 2. (2𝑥2𝑦2)5 =

3. (42𝑥3𝑦𝑧2)3 = 4. (2𝑥2𝑦3)2(2𝑥𝑦)4 =

Quotient to a Power: If you have a quotient of multiple exponential expressions all being raised to a

power, usually using parentheses, you should ____________________ all of the inside exponents by the

outside exponent.

• It is basically identical to the product to a power rule with one exception- if you can reduce

the quotient using the quotient rule before distributing, you need to.

• All of the other additional mentions for the product to a power rule also apply to this rule.

Why? Remember than an exponent tells us how many times to multiply a base to itself. Using that

logic, rewrite and simplify the following exponential expression:

(𝑥3

𝑦2)

3

=

Examples:

5. (2𝑥2𝑦2

𝑧3)4

= 6. (48𝑥20𝑦30𝑧15

16𝑥18𝑦35𝑧15)8

=

7. (−2𝑥

𝑦)2