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Mathematics

Grade 9

Numbers

Module 2

Easier algebra with exponents

Easier algebra with exponents

CLASS WORK

  • Do you remember how exponents work? Write down the meaning of “three to the power seven”. What is the base? What is the exponent? Can you explain clearly what a power is?
  • In this section you will find many numerical examples; use your calculator to work through them to develop confidence in the methods.

1 DEFINITION

2 3 = 2 × 2 × 2 and a 4 = a × a × a × a and b × b × b = b 3

also

(a+b) 3 = (a+b) × (a+b) × (a+b) and 2 3 4 = 2 3 × 2 3 × 2 3 × 2 3 size 12{ left ( { {2} over {3} } right ) rSup { size 8{4} } = left ( { {2} over {3} } right ) times left ( { {2} over {3} } right ) times left ( { {2} over {3} } right ) times left ( { {2} over {3} } right )} {}

1.1 Write the following expressions in expanded form:

4 3 ; (p+2) 5 ; a 1 ; (0,5) 7 ; b 2 × b 3 ;

1.2 Write these expressions as powers:

7 × 7 × 7 × 7

y × y × y × y × y

–2 × –2 × –2

(x+y) × (x+y) × (x+y) × (x+y)

1.3 Answer without calculating: Is (–7) 6 the same as –7 6 ?

  • Now use your calculator to check whether they are the same.
  • Compare the following pairs, but first guess the answer before using your calculator to see how good your estimate was.

–5 2 and (–5) 2 –12 5 and (–12) 5 –1 3 and (–1) 3

  • By now you should have a good idea how brackets influence your calculations – write it down carefully to help you remember to use it when the problems become harder.
  • The definition is:

a r = a × a × a × a × . . . (There must be r a’s, and r must be a natural number)

  • It is good time to start memorising the most useful powers:

2 2 = 4; 2 3 = 8; 2 4 = 16; etc. 3 2 = 9; 3 3 = 27; 3 4 = 81; etc. 4 2 = 16; 4 3 = 64; etc.

Most problems with exponents have to be done without a calculator!

2 MULTIPLICATION

  • Do you remember that g 3 × g 8 = g 11 ? Important words: multiply ; same base

2.1 Simplify: (don’t use expanded form)

7 7 × 7 7

(–2) 4 × (–2) 13

( ½ ) 1 × ( ½ ) 2 × ( ½ ) 3

(a+b) a × (a+b) b

  • We multiply powers with the same base according to this rule:

a x × a y = a x+y also a x + y = a x × a y = a y × a x size 12{a rSup { size 8{x+y} } =a rSup { size 8{x} } times a rSup { size 8{y} } =a rSup { size 8{y} } times a rSup { size 8{x} } } {} , e.g. 8 14 = 8 4 × 8 10 size 12{8 rSup { size 8{"14"} } =8 rSup { size 8{4} } times 8 rSup { size 8{"10"} } } {}

3 DIVISION

  • 4 6 4 2 = 4 6 2 = 4 4 size 12{ { {4 rSup { size 8{6} } } over {4 rSup { size 8{2} } } } =4 rSup { size 8{6 - 2} } =4 rSup { size 8{4} } } {} is how it works. Important words: divide ; same base

3.1 Try these: a 6 a y size 12{ { {a rSup { size 8{6} } } over {a rSup { size 8{y} } } } } {} ; 3 23 3 21 size 12{ { {3 rSup { size 8{"23"} } } over {3 rSup { size 8{"21"} } } } } {} ; a + b p a + b 12 size 12{ { { left (a+b right ) rSup { size 8{p} } } over { left (a+b right ) rSup { size 8{"12"} } } } } {} ; a 7 a 7 size 12{ { {a rSup { size 8{7} } } over {a rSup { size 8{7} } } } } {}

  • The rule for dividing powers is: a x a y = a x y size 12{ { {a rSup { size 8{x} } } over {a rSup { size 8{y} } } } =a rSup { size 8{x - y} } } {} .

Also a x y = a x a y size 12{a rSup { size 8{x - y} } = { {a rSup { size 8{x} } } over {a rSup { size 8{y} } } } } {} , e.g. a 7 = a 20 a 13 size 12{a rSup { size 8{7} } = { {a rSup { size 8{"20"} } } over {a rSup { size 8{"13"} } } } } {}

4 RAISING A POWER TO A POWER

  • e.g. 3 2 4 size 12{ left (3 rSup { size 8{2} } right ) rSup { size 8{4} } } {} = 3 2 × 4 size 12{3 rSup { size 8{2 times 4} } } {} = 3 8 size 12{3 rSup { size 8{8} } } {} .

4.1 Do the following:

  • This is the rule: a x y = a xy size 12{ left (a rSup { size 8{x} } right ) rSup { size 8{y} } =a rSup { size 8{ ital "xy"} } } {} also a xy = a x y = a y x size 12{a rSup { size 8{ ital "xy"} } = left (a rSup { size 8{x} } right ) rSup { size 8{y} } = left (a rSup { size 8{y} } right ) rSup { size 8{x} } } {} , e.g. 6 18 = 6 6 3 size 12{6 rSup { size 8{"18"} } = left (6 rSup { size 8{6} } right ) rSup { size 8{3} } } {}

5 THE POWER OF A PRODUCT

  • This is how it works:

(2a) 3 = (2a) × (2a) × (2a) = 2 × a × 2 × a × 2 × a = 2 × 2 × 2 × a × a × a = 8a 3

  • It is usually done in two steps, like this: (2a) 3 = 2 3 × a 3 = 8a 3

5.1 Do these yourself: (4x) 2 ; (ab) 6 ; (3 × 2) 4 ; ( ½ x) 2 ; (a 2 b 3 ) 2

  • It must be clear to you that the exponent belongs to each factor in the brackets.
  • The rule: (ab) x = a x b x also a p × b p = ab b size 12{a rSup { size 8{p} } times b rSup { size 8{p} } = left ( ital "ab" right ) rSup { size 8{b} } } {} e.g. 14 3 = 2 × 7 3 = 2 3 7 3 size 12{"14" rSup { size 8{3} } = left (2 times 7 right ) rSup { size 8{3} } =2 rSup { size 8{3} } 7 rSup { size 8{3} } } {} and 3 2 × 4 2 = 3 × 4 2 = 12 2 size 12{3 rSup { size 8{2} } times 4 rSup { size 8{2} } = left (3 times 4 right ) rSup { size 8{2} } ="12" rSup { size 8{2} } } {}

6 A POWER OF A FRACTION

  • This is much the same as the power of a product. a b 3 = a 3 b 3 size 12{ left ( { {a} over {b} } right ) rSup { size 8{3} } = { {a rSup { size 8{3} } } over {b rSup { size 8{3} } } } } {}

6.1 Do these, but be careful: 2 3 p size 12{ left ( { {2} over {3} } right ) rSup { size 8{p} } } {} 2 2 3 size 12{ left ( { { left ( - 2 right )} over {2} } right ) rSup { size 8{3} } } {} x 2 y 3 2 size 12{ left ( { {x rSup { size 8{2} } } over {y rSup { size 8{3} } } } right ) rSup { size 8{2} } } {} a x b y 2 size 12{ left ( { {a rSup { size 8{ - x} } } over {b rSup { size 8{ - y} } } } right ) rSup { size 8{ - 2} } } {}

  • Again, the exponent belongs to both the numerator and the denominator.
  • The rule: a b m = a m b m size 12{ left ( { {a} over {b} } right ) rSup { size 8{m} } = { {a rSup { size 8{m} } } over {b rSup { size 8{m} } } } } {} and a m b m = a b m size 12{ { {a rSup { size 8{m} } } over {b rSup { size 8{m} } } } = left ( { {a} over {b} } right ) rSup { size 8{m} } } {} e.g. 2 3 3 = 2 3 3 3 = 8 27 size 12{ left ( { {2} over {3} } right ) rSup { size 8{3} } = { {2 rSup { size 8{3} } } over {3 rSup { size 8{3} } } } = { {8} over {"27"} } } {} and a 2x b x = a 2 x b x = a 2 b x size 12{ { {a rSup { size 8{2x} } } over {b rSup { size 8{x} } } } = { { left (a rSup { size 8{2} } right ) rSup { size 8{x} } } over {b rSup { size 8{x} } } } = left ( { {a rSup { size 8{2} } } over {b} } right ) rSup { size 8{x} } } {}

end of CLASS WORK

TUTORIAL

  • Apply the rules together to simplify these expressions without a calculator.

1. a 5 × a 7 a × a 8 size 12{ { {a rSup { size 8{5} } times a rSup { size 8{7} } } over {a times a rSup { size 8{8} } } } } {} 2. x 3 × y 4 × x 2 y 5 x 4 y 8 size 12{ { {x rSup { size 8{3} } times y rSup { size 8{4} } times x rSup { size 8{2} } y rSup { size 8{5} } } over {x rSup { size 8{4} } y rSup { size 8{8} } } } } {}

3. a 2 b 3 c 2 × ac 2 2 × bc 2 size 12{ left (a rSup { size 8{2} } b rSup { size 8{3} } c right ) rSup { size 8{2} } times left ( ital "ac" rSup { size 8{2} } right ) rSup { size 8{2} } times left ( ital "bc" right ) rSup { size 8{2} } } {} 4. a 3 × b 2 × a 3 a × b 5 b 4 × ab 3 size 12{a rSup { size 8{3} } times b rSup { size 8{2} } times { {a rSup { size 8{3} } } over {a} } times { {b rSup { size 8{5} } } over {b rSup { size 8{4} } } } times left ( ital "ab" right ) rSup { size 8{3} } } {}

5. 2 xy × 2x 2 y 4 2 × x 2 y 3 2 xy 3 size 12{ left (2 ital "xy" right ) times left (2x rSup { size 8{2} } y rSup { size 8{4} } right ) rSup { size 8{2} } times left ( { { left (x rSup { size 8{2} } y right ) rSup { size 8{3} } } over { left (2 ital "xy" right ) rSup { size 8{3} } } } right )} {} 6. 2 3 × 2 2 × 2 7 8 × 4 × 8 × 2 × 8 size 12{ { {2 rSup { size 8{3} } times 2 rSup { size 8{2} } times 2 rSup { size 8{7} } } over {8 times 4 times 8 times 2 times 8} } } {}

end of TUTORIAL

Some more rules

CLASS WORK

1 Consider this case: a 5 a 3 = a 5 3 = a 2 size 12{ { {a rSup { size 8{5} } } over {a rSup { size 8{3} } } } =a rSup { size 8{5 - 3} } =a rSup { size 8{2} } } {}

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Mathematics grade 9. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col11056/1.1
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