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

 

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

Check your results in the following window.

Index 1 is the exponent of the first factor; Index 2 is the exponent of the second factor. Increase the number of decimal places if necessary.


10. Work out the following and write the answers in your notebook in index form: 

a) 2 * 24 * 25    b) 42 * 44 * 43
c) 8 * 8 * 84

d) 2 * 2-4 * 25    e) 4-2 * 44 * 4-3
f) 8-1 * 8 * 84

Check your results in the following window.

Index 1 is the exponent of the first factor; Index 2 is the exponent of the second factor and Index 3 is the exponent of the third.


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

 

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

Check your results in the following window.

Index 1 is the exponent of the numerator; Index 2 is the exponent of the denominator.


Powers and products

To work out (2*3)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

 

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

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

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

 

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

Work out the answers in your notebook and check them in the following window.


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  Fernando Arias Fernández-Pérez
 
Spanish Ministry of Education. Year 2001
 
 

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