Powers: fractional indices.
3rd year of secondary education.
 

The root of a number.

We know that 72 = 49. We can also express this in the following way:

Therefore, 7 is equal to the square root of 49. In general the square root of a number a can be expressed in terms of b as b2 = a.

In the same way the nth root of a number a is expressed in terms of b as bn = a.

So we can say:

where number a is called the radicand and number n the index.

For example:

It is worth pointing out that not all numbers have roots. There is no square root for the number -4 as the square of any number, be it positive or negative, is always positive. This is also the reason why neither the square root of a negative number nor the root of an even index of a negative number do not exist.

14. Use the table of perfect squares and cubes as a point of reference to find the following roots:

Check your answers in the following window.


Powers with fractional indices.

If we look back at the first example above we can say that:

as, referring back to the rule to work out a power raised to another power:

(81/3)3 = 81/3 * 3 = 81 = 8

In general, we can say that:

as

(a1/n)n = a1/n * n = a1 = a

This can also be expressed as:

15. Express the following numbers raised to fractional powers in root form. Work out the value of the number and use the following window to check your results. Increase the number of decimal places if necessary.

a) 163/4   b) 272/3    c) 1254/3

d) 645/6   e) 100-3/2   f) 8-2/3

Check your results in the following window.


Rules of numbers raised to fractional powers.

Numbers raised to fractional powers follow the same rules as those numbers raised to whole numbers. Let's go over these rules again one by one:

The product of two powers with the same base number.

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

This rule works for any base number and for indices, m and n, regardless of whether they are positive, negative, whole numbers or fractions.

16. Express the following products in your exercise book in index form:

a) 23/5 * 27/2
b) 35/2 * 32/3
c) 52/5 * 52/3
d) 2-3/10 * 22/5
e) 3-5/2 * 3-2/3
f) 10-1/5 * 101/3

Check your results in the following window.

Dividing two powers with the same base number.

You can work out the general rule, which is true for both positive and negative powers, 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

17. Express the following divisions in index form:

a) 27/3 : 24/3
b) 31/5 : 32/3
c) 51/6 : 51/3
d) 643/2 : 64-1/3
e) 3-1/2 : 33/2
f) 8-4/3 : 8-5/3

Check your results in the following window.

Powers and products

A product raised to a power is equal to multiplying these numbers raised to the same power

 

(a*b)m = am * bm

18. Express the following as the product of two numbers, each raised to a power (in product form):

a) (2*5)1/6
b) (3*4)3/2
c) (2*8)2/3
d) (4*6)3/4
e) (2*5)-1/2
f) (3*2)-2/3
g) (2*5)-5/3

Check your results in the following window.

Powers and division.

Likewise, we can deduce that:

A division raised to a power is equal to dividing these numbers raised to the same power

(a/b)m = am / bm

19. Express the following as the division of two numbers, each raised to a power:

a) (18/2)5/6
b) (64/4)1/2
c) (75/5)2/3
d) (12/3)3/4
e) (18/2)-2/3
f) (32/4)-3/2
g) (81/27)-1/3
h) (32/9)-1/4

Check your results in the following window.

Raising a power to another power.

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

20. Work out the following in your exercise book, expressing each example as a number raised to one power:

a) (21/3)7
b) (35)1/3
c) (51/5)1/3
d) (2-3/2)4
e) (33/4)-1/4
f) (25-1/2)-3

Check your results in the following window.

21. Use the examples above to write a list of all the rules governing roots and their operations.


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