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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} } } {}

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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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