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
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:
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+yalso
, e.g.
3 DIVISION
is how it works. Important words:
divide ;
same base
3.1 Try these:
;
;
;
The rule for dividing powers is:
.
Also
, e.g.
4 RAISING A POWER TO A POWER
e.g.
=
=
.
4.1 Do the following:
This is the rule:
also
, e.g.
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
xalso
e.g.
and
6 A POWER OF A FRACTION
This is much the same as the power of a product.
6.1 Do these, but be careful:
Again, the exponent belongs to both the numerator and the denominator.
The rule:
and
e.g.
and
end of CLASS WORK
TUTORIAL
Apply the rules together to simplify these expressions without a calculator.
1.
2.
3.
4.
5.
6.
end of TUTORIAL
Some more rules
CLASS WORK
1 Consider this case:
Questions & Answers
A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
please, I'm a physics student and I need help in physics
Adjanou
chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.
Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?