0:00

Let's look at the properties of integer exponents.

[SOUND] For n and m integers and a and b real numbers, we have the following

properties. The first property states that a^n * a^m

= a^n+m. And it's often referred to as the product

rule. For example, a^2 * a^3 = a^2+3 or a^5.

The second property states the (a^n)^m = a^n*m and is often referred to as the

power of a power rule. For example (a^2)^3 = a^2*3 or a^6.

Notice that in the first property when the bases is same we add the exponents.

Whereas in the second property we have a power of a power, we multiply the

exponents. These properties are often confused.

So know when to add, and when to multiply.

The third property states that (ab)^m = a^m * b^m, and is often referred to as

the power of a product rule. For example, a * b^2 = a^2 * b^2.

2:05

= a^m / b^m and is often referred to as the power of a quotient rule.

For example, (a/b)^3 = a^3 / b^3. And here, we're assuming, of course, that

b is not equal to zero. Alright the fifth property states that

a^m / a^n = a^m-n. And is often referred to as the quotient

rule. For example, a^5 divided by a^2 = a^5-2

2:50

or a^3. So when the bases are the same, and we're

dividing, we subtract the exponents. The sixth property states that a^0 = 1,

for a not equal to 0. 0^0 is not defined for various reasons.

This is often referred to as the zero exponent rule.

For example, 3^0 = 1. And the last property to consider here

[SOUND] is that a^-n = 1 / a^+n and is often referred to as a negative exponent

rule. For example a^-3 = 1 / a^3.

3:48

Alright let's see an example [SOUND] Let's simplify the following expression

and write our answer using only positive exponents.

Since multiplication is both commutative and associative we can regroup this

multiplication as follows. We can take all the numbers first, the 2,

the 3, and the 5, and multiply them, so this is equal to 2 * 3 * 5 and then

multiply the w terms together, so times w^-5 * 2^4.

And then group the v terms together. We have this term and this term.

So, * v^-6 * v^7. And finally we'll group the u terms

together. So * u^7 * u^2.

4:58

So this is equal to 2 * 3 * 5 is 30 and then * w^-5+4 by the product rule because

these bases are the same, we can add those exponents.

Same with the v terms. So it'll be v^-6+7 and finally we'll do

the same with the u terms. So it's u^7 + 2 which is = 30 * w^-1 * v^

= 1 * u^9. And then by the negative exponent rule,

this is = 30 * 1 / w^1. Remember, we want to write our answer

using only positive exponents, and when the exponent of a variable is 1, we

usually do not write it, so writing this as 1 fraction and dropping those

exponents of 1 gives us our answer of 30v, u^9 / w.

Alright, let's use another example. Let's simplify this expression and write

our answer using only positive exponents. Well, the first thing we can do is

simplify what's inside these parenthesis by again grouping like terms.

So this is equal to, let's group our numbers together so 6 / 3 and then times

grouping our m terms together. We have m / m^-1 and then finally

grouping the n terms together, we have (n^-2/n^2)^-3 which is equal to, 6 / 3 =

2 and then * m^1 - 1 - and this comes from the quotient rule because the bases

are the same we subtract the exponents. And we'll do the same with the n terms,

so it's (n^-2 - 2)^-3. And this is equal to 2m^2 because that's

(1 - 1 - * n^-4)^-3. And then, by the power of a product rule,

we can raise each of the factors to the -3rd power.

Which = 1 / 2 ^ +3 by our negative exponent rule.

And then times, we have a power of a power, so remember we multiply 2 * -3

which is -6. Same with the n term we have a power of a

power, so we multiply. So we have -4 * -3 which is +12.

And remember we want to write our answer using only positive, exponent.

So let's use that negative exponent rule again on this m term.

So this is equal to, we have 1 / 2 ^ 3 = 8.

And then we have 1 / m ^ +6, n ^ 12 And writing it as one fraction will give us

our answer of n ^ 12 / 8 * m ^ 6. And this is how we work with integer

exponents. Thank you, and we'll see you next time.

[SOUND]