Lesson 13, Section 2
Three Rules of Exponents
Rule 1. Same base
aman = am + n
"To multiply powers of the same base, add the exponents."
For example, a2a3 = a5.
Why do we add the exponents? Because of what the symbols mean. Section 1.
Example 1. Multiply 3x2· 4x5· 2x
Solution. The problem means (Lesson 5): Multiply the numbers, then combine the powers of x :
3x2· 4x5· 2x = 24x8
Two factors of x -- x2 -- times five factors of x -- x5 -- times one factor of x, produce a total of 2 + 5 + 1 = 8 factors of x : x8.
Problem 1. Multiply. Apply the rule Same Base.
To see the answer, pass your mouse over the colored area.
Problem 2. Distinguish the following:
x· x and x + x.
x· x = x². x + x = 2x.
Example 2. Compare the following:
a) x· x5 b) 2· 25
a) x· x5 = x6
b) 2· 25 = 26
Part b) has the same form as part a). It is part a) with x = 2.
One factor of 2 multiplies five factors of 2 producing six factors of 2.
2· 2 = 4 is not correct here.
Problem 3. Apply the rule Same Base.
Problem 4. Apply the rule Same Base.
Rule 2: Power of a product of factors
(ab)n = anbn
"Raise each factor to that same power."
For example, (ab)3 = a3b3.
Why may we do that? Again, according to what the symbols mean:
(ab)3 = ab· ab· ab = aaabbb = a3b3.
The order of the factors does not matter:
ab· ab· ab = aaabbb.
Problem 5. Apply the rules of exponents.
Rule 3: Power of a power
(am)n = amn
"To take a power of a power, multiply the exponents."
For example, (a2)3 = a2· 3 = a6.
Why do we do that? Again, because of what the symbols mean:
(a2)3 = a2a2a2 = a3· 2 = a6
Problem 6. Apply the rules of exponents.
Example 3. Apply the rules of exponents: (2x3y4)5
Solution. Within the parentheses there are three factors: 2, x3, and y4. According to Rule 2 we must take the fifth power of each one. But to take a power of a power, we multiply the exponents. Therefore,
(2x3y4)5 = 25x15y20
Problem 7. Apply the rules of exponents.
Problem 8. Apply the rules of exponents.
a) 2x5y4(2x3y6)5 = 2x5y4· 25x15y30 = 26x20y34
b) abc9(a2b3c4)8 = abc9· a16b24c32 = a17b25c41
Problem 9. Use the rules of exponents to calculate the following.
a) (2· 10)4 = 24· 104 = 16· 10,000 = 160,000
b) (4· 102)3 = 43· 106 = 64,000,000
c) (9· 104)2 = 81· 108 = 8,100,000,000
Example 4. Square x4.
Solution. (x4)2 = x8.
Thus to square a power, double the exponent.
Problem 10. Square the following.
Part c) illstrates: The square of a number is always positive.
(−6)(−6) = +36. The Rule of Signs.
Except 02 = 0.
Problem 11. Apply a rule of exponents -- if possible.
In summary: Add the exponents when the same base appears twice: x2x4 = x6. Multiply the exponents when the base appears once -- and in parentheses: (x2)5 = x10.
Problem 12. Apply the rules of exponents.
Problem 13. Apply a rule of exponents or add like terms -- if possible.
a) 2x2 + 3x4 Not possible. These are not like terms. Lesson 1.
b) 2x2· 3x4 = 6x6. Rule 1.
c) 2x3 + 3x3 = 5x3. Like terms. The exponent does not change.
d) x2 + y2 Not possible. These are not like terms.
e) x2 + x2 = 2x2. Like terms.
f) x2· x2 = x4. Rule 1
g) x2· y3 Not possible. Different bases.
h) 2· 26 = 27. Rule 1
i) 35 + 35 + 35 = 3· 35 (On adding those like terms) = 36.
We will continue the rules of exponents in Lesson 21.
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