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Let's look at the properties of integer exponents.

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

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

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

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

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

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

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

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

Â exponents. These properties are often confused.

Â So know when to add, and when to multiply.

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

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

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

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

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

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

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

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

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

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

Â This is often referred to as the zero exponent rule.

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

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

Â rule. For example a^-3 = 1 / a^3.

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

Â and write our answer using only positive exponents.

Â Since multiplication is both commutative and associative we can regroup this

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

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

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

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

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

Â together. So * u^7 * u^2.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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