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Authoritative PPT Summary
"Power of product and power of power" PPT courseware 3
Review & Thoughts
The meaning of power: a·a· … ·a=an
Rules for multiplication of powers with the same base: am · an=am+n (m, n are both positive integers)
The power arithmetic rule: (am)n=amn (m, n are both positive integers)
A little test of your skills:
Calculation using powers raised
(62)4 (a2)3
(am)2 (am)n
Rules for multiplication of powers:
When raising powers, the base remains unchanged and the exponents are multiplied.
Student exercises:
1. (102) 3 (b5) 5 (an) 3 ─ (x2) m
2. (y2) 3. y2. 2 (a2) 6. a3 -- (a3) 4. a3
3. (-32) 3. (-33) 2 (-x) 2. (-x) 3
4. ① (4·2n)·(4·2n) is equal to
( )A. 4·2n B. 42n+4 C. 22n D． 22n+4
② The number of correct propositions in the following calculations is ( )
1. am·a2=a2m 2． (a3)2=a5
4. x3·x2=x6 3. (-a3 )2.a4 = a9
(A) 1 (B) 2 (C) 3 (D) None of the above answers are correct
Explore & communicate
(1) According to the definition of power (the meaning of power), what does (ab)3 represent?
(2) In order to calculate (simplify) the expression ab·ab·ab, the commutative law and associative law of multiplication can be applied.
In what form can it be written?
(3) Starting from the special (ab)3=a3b3, can you think of a general formula?
(ab)3=ab·ab·ab
=a·a·a·b·b·b
=a3·b3
multiplication rule for products
(ab)n = an·bn (m,n are both positive integers)
The above formula shows:
The product when each factor is raised to the power of a product =_____________.
Can you tell the meaning of the word "factor" in the law?
(a+b)n, can it be calculated using the multiplication rule of products?
That is, is "(a+b)n= an·bn" true?
Also, is "(a+b)n= an+an" true?
(ab)n = an·bn (m,n are both positive integers)
Use in reverse: an·bn = (ab)n
Try a simple method to calculate:
(1) 23×53= (2×5)3= 103
(2) 28×58= (2×5)8= 108
(3) (-5)16 × (-2)15= (-5)×[(-5)×(-2)]15= -5×1015;
(4) 24 × 44 ×(-0.125)4 = [2×4×(-0.125)]4
= 14
= 1.
summary
The meaning of power: a·a· … ·a=an
Rules for multiplication of powers with the same base: am · an=am+n
The exponentiation rule: (ab)n=ambn
The product raised to the power = the product of each factor raised to the power.
Some calculations can be simplified by using am · an =am+n, (am)n =amn in reverse.
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