Powers and roots: operations and rules.
3rd year of secondary education.
 
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

8. Calculate the following in index form and write your answers in your exercise book:

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.

9. Work out the following and write the answers in your exercise book 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 is 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

 

10. Calculate the following and write the answers in index form in your exercise book:

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)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

 

11. 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 exercise book 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

12. 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 exercise book and check them in the following window. 

Raising a power to another 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

13. Work out the following in your exercise book, expressing 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 exercise book and check them in the following window.

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