x - 2
x-1
m.
= X
n.
x-1
x +2
2 x + 1
2x + 5
+
2
4
4.
Answers
Answer:
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.
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Do the problem yourself first!
a) 5x2 · 6x4 = 30x6 b) 7x3 · 8x6 = 56x9
c) x · 5x4 = 5x5 d) 2x · 3x · 4x = 24x3
e) x3 · 3x2 · 5x = 15x6 f) x5 · 6x8y2 = 6x13y2
g) 4x · y · 5x2 · y3 = 20x3y4 h) 2xy · 9x3y5 = 18x4y6
i) a2b3a3b4 = a5b7 j) a2bc3b2ac = a3b3c4
k) xmynxpyq = xm + pyn+ q l) apbqab = ap + 1bq + 1
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
Solution.
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.
a) xx7 = x8 b) 3 · 37 = 38 c) 2 · 24 · 25 = 210
d) 10 · 105 = 106 e) 3x · 36x6 = 37x7
Problem 4. Apply the rule Same Base.
a) xnx2 = xn + 2 b) xnx = xn + 1
c) xnxn = x2n d) xnx1 − n = x
e) x · 2xn − 1 = 2xn f) xnxm = xn + m
g) x2nx2 − n = xn + 2
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.