MAT/117
Week 5 Discussion Question 1
Review section 10.2 (p. 692) of your text. Describe two laws of exponents and provide an
example illustrating each law. Explain how to simplify your expression. How do the laws work
with rational exponents? Provide the class with a third expression to simplify that includes
rational (fractional) exponents.
Product Law is to raise a product to a power, (we can raise each factor to the power).
With two
powers of the same base like (x^5)(x^6); how do you simplify that? Just remember that you’re
counting factors.
x^5 = (x)(x)(x)(x)(x) and x^6 = (x)(x)(x)(x)(x)(x)
Now multiply them together:
(x5)(x6) = (x)(x)(x)(x)(x)(x)(x)(x)(x)(x)(x) = x^11
Whenever you multiply any two powers of the same base, you end up with a number of factors
equal to the total of the two powers? In other words, when the bases are the same, you find the
new power by just adding the exponents. Now with the powers of different bases be careful the
rule above works only when multiplying powers of the same base. For instance,
(x^3)(y^4) = (x)(x)(x)(y)(y)(y)(y)
If you write out the powers, you see there’s no way you can combine them.
If the bases are different but the exponents are the same, then you can combine them.
Example:
(x³)(y³) = (x)(x)(x)(y)(y)(y)
But you know that it doesn’t matter what order you do your multiplications in or how you group
them. Therefore,
(x)(x)(x)(y)(y)(y) = (x)(y)(x)(y)(x)(y) = (xy)(xy)(xy)
But from the very definition of powers, you know that’s the same as (xy)³. And it works for any
common power of two different bases: All the laws of exponents work in both directions. If you
see (4x)³ you can decompose it to (4³)(x³), and if you see (4³)(x³) you can combine it as (4x)³.
Example:
3^7 * 3^2 = 3^(7 + 2) = 3^9
Problem for class:
5^1/2 * 5^1/4 = ?
Quotient Law is to raise a quotient to a power, (we can raise both the numerator and the
denominator to the power. The power of a quotient may be developed from the following
example:
(2/3)^3 = 2/3*2/3*2/3
= 2*2*2
3*3*3
= 2^3/3^3
Therefore,
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View Full Document(2/3)^3 = 2^3/3^3
The law is stated as follows: The power of a quotient is equal to the quotient obtained when the
dividend and divisor are each raised to the indicated power separately, before the division is
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 Spring '10
 a.b.
 Division, Exponents, Equivalence relation, Power of two, Mathematics in medieval Islam

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