powers and indices
DESCRIPTION
Powers and Indices. Slideshow 10, Mathematics Mr Richard Sasaki, Room 307. Objectives. To recall simple algebraic rules To learn how products of an unknown make a power To learn how to multiply and divide powers of an unknown. Review. - PowerPoint PPT PresentationTRANSCRIPT
Slideshow 10, MathematicsMr Richard Sasaki, Room 307
Powers and Indices
Objectives• To recall algebraic rules learned so far• To learn how products of an unknown
make a power• To learn how to multiply and divide
powers of an unknown
ReviewLet’s review the main rules we have learned so far.6×𝑥¿6 𝑥𝑥×𝑥¿𝑥2𝑥× 𝑦¿𝑥𝑦𝑥+𝑥¿2 𝑥𝑥+𝑦¿𝑥+𝑦
−5×𝑥¿−5 𝑥𝑥−𝑥¿0𝑥÷ 𝑦¿𝑥𝑦𝑥÷ 𝑥¿1
Also, writing expressions in alphabetical order is usually preferred () but not crucial. ( is fine).
Powers (Indices)As we know, . 𝑥2We call or .-squared to the power 2
The small 2 symbol at the top is called the power or index.Note: Power and Index mean the same thing. Indices is plural of index in this context.
How about ? 𝑥×𝑥×𝑥=𝑥3We call or .-cubed to the power 3How about ? 𝑥×𝑥×𝑥×𝑥=𝑥4
We call . to the power 4
Note: onwards are read “to the power” as well.
Calculation (Multiplication)What do you think is? 𝑥1=𝑥Just one is present.
Let’s try some multiplication.
ExampleCalculate . Have a guess!
𝑎4×𝑎3=¿(𝑎×𝑎×𝑎×𝑎)×(𝑎×𝑎×𝑎)¿𝑎7
So… .𝑥𝑎+𝑏
What will happen when we divide indices?
Note: Powers is one area where we see and symbols in algebra (before simplified).
Calculation (Division)ExampleCalculate .
𝑎6÷𝑎3=¿𝑎×𝑎×𝑎×𝑎×𝑎×𝑎
𝑎×𝑎×𝑎¿𝑎×𝑎×𝑎¿𝑎3
So… .𝑥𝑎−𝑏
What do you think might equal? Have a think!
Answers𝑥3 𝑦 5 𝑥5𝑎7 𝑥3 𝑦 6
𝑥2 𝑦 3 𝑥𝑎6 𝑎 𝑥6
𝑎5 𝑎7 𝑥6
𝑥3 𝑦 7 𝑥12
𝑥9 𝑦 4 𝑦 5
𝑎9 or
1
Negative Powers and Zero and . So , right? Why?
𝑥3÷ 𝑥5𝑥3−5𝑥− 2
𝑥3
𝑥5𝑥×𝑥×𝑥
𝑥×𝑥×𝑥×𝑥×𝑥1
𝑥2
So… .1
𝑥𝑎 Writing this in both ways is fine.
Why does ?
𝑥0=¿𝑥1÷𝑥1=¿𝑥÷ 𝑥=¿1Note: Any number to the power zero is 1.
Answers1𝑥
1
𝑦31
𝑦31
𝑎5
11 1 1
𝑥− 1 𝑥− 5 𝑥− 3 𝑥− 5
1
𝑥21
𝑎4
1
𝑦41𝑥
2𝑥
3
𝑦32
𝑎27
𝑥33
𝑎2𝑥2
𝑦
Brackets and Other CalculationsHow would we calculate ?
(𝑥2 )3=¿𝑥2×𝑥2×𝑥2=¿𝑥6So… .𝑥𝑎𝑏
Be careful! (usually).ExampleCalculate .
4 (𝑎2 )3×2𝑎2=¿4× (𝑎2 )3×2×𝑎2¿8×𝑎6×𝑎2¿8 𝑎8
Answers - Easy𝑥7 𝑥3 𝑥121 𝑎 𝑎2 𝑥 5 𝑦 3 52 3 𝑥2𝑦 2
0 𝑥6 𝑦 𝑥2
4 𝑥2 16 𝑦2 4 𝑥4
2 𝑥2 𝑎2𝑏2 6 𝑥3
2 𝑥3 6 𝑥3 𝑦 2 𝑥2 𝑦3
𝑥 1 𝑥− 3
𝑥 2 3
Answers - Hard
2 𝑥− 2 3 𝑦−3 3𝑎−2 2𝑎−3
0 8 𝑥6 21 𝑥6 𝑦6 𝑥7
3 𝑥 𝑦2 4 𝑥2 𝑦 2 3 𝑥2
𝑥 𝑦 312 2 𝑥6 𝑦3
8 𝑥6 𝑦3 256 𝑥12 2 𝑥3𝑥2 𝑥9 𝑦3 𝑥4 𝑥𝑦 96 𝑥5 𝑦10