Powers: operations and rules. | |
4th year of secondary education. Option A. | |
Multiplying powers with the same base number. | |
If
we want to multiply two powers with the same base number, e.g. 43 * 45
we do the following:
43 = 4 * 4 * 4
and
45 = 4 * 4 * 4 * 4 * 4,
so
43 * 45 = (4 * 4 * 4) * (4 * 4 * 4
* 4 * 4) = 48 = 43+5
In general:
The product of two powers with the same
base number is the same base number whose index is the sum of the other two
indices.
am * an
= am+n
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9. Work out the following in index form and write your answers in your notebook:
a) 23 * 27
b) 35 * 33
; c) 55 * 53
d) 2-3 * 25
e) 3-5 * 3-3
; f) 5-5 * 53
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Check your results in the following window. | |
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10. Work out the following and write the answers in your notebook in index form:
a) 2 * 24
* 25
b) 42 * 44
* 43
d) 2 * 2-4
* 25
e) 4-2 * 44
* 4-3
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Check your results in the following window. | |
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Dividing powers with the same base number. | |
You can work out the
general rule in the same way as you did to find the product:
Dividing two powers with the same base number gives the same base number whose index is the difference between the other two indices. am : an = am-n For example: 45 : 43 = (4 * 4 * 4 * 4 * 4) : (4 * 4 * 4) = 42 = 45-3
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11. Calculate the following and write the answers in index form
in your notebook:
a) 27 : 23
b) 35 : 33
c) 56 : 53
d) 27 : 2-3
e) 3-2 : 32
f) 5-4 : 5-3
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Check your results in the following window. | |
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Powers and products | |
To
work out (2*3)2 we have to do the following:
(2*3)3 = (2*3) * (2*3) * (2*3) = (2*2*2)
* (3*3*3) = 23 * 33
To calculate the result we have to multiply 2*3 and cube the
product: (2*3)3 = 63 = 216
Or, we can cube each of the factors, 23 = 8 and
33= 27, and multiply the result: 8*27 = 216.
In general: A product raised to a power is equal to multiplying these numbers raised to the same power (a*b)m = am * bm
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12. Express the following in product form:
a) (2*5)6
b) (3*4)2
c) (2*8)3
d) (4*6)4
e) (2*5)-2
f) (3*2)-3
g) (2*5)-3
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Work out the answers in your notebook and check them in the following window. |
Powers and division. | |
Likewise, we can easily deduce that: A division raised to a power is equal to dividing these numbers raised to the same power (a/b)m = am / bm | |
13. Express the following in division form:
a) (18/2)6
b) (8/4)2
c) (10/5)3
d) (12/3)4
e) (18/2)-3
f) (8/4)-2
g) (10/5)-3
h) (9/3)-4
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Work out the answers in your notebook and check them in the following window. |
Raising a power to a power | |
If
we want to work out (45)3 we have to do the following:
(45)3 = 45 * 45 * 45 =
45+5+5 = 45*3
Therefore we can deduce the following rule:
A power raised to another power is the
same as the base number raised to the product of these two powers:
(am)n = am*n
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14. Simplify the following in your notebook and express each example as a number
raised to one power:
a) (23)7
b) (35)3
c) (55)3
d) (2-3)2
e) (33)-2
f) (5-2)-3
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Work out the answers in your notebook and check them in the following window. |
Fernando Arias Fernández-Pérez | ||
Spanish Ministry of Education. Year 2001 | ||
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