Powers: fractional indices. | |
4th year of secondary education. Option A. | |
The root of a number. | |
We know that 72 = 49. We can also express this in the following way:
We say that 7 is equal to the square root of 49. In general,
the square root of a number a is defined as another number b
where b2 = a.
In the same way, we can define the nth root of
a number a as b where bn
= a
and we write:
The number a is called the radicand and n
is the index.
For
example,
It is important to be aware
that not all numbers have roots. For example, -4 does not have a square root as
the square of any number, whether it be positive or negative, is always
positive. For this reason no negative number has a square root and no negative
number has a root with an even index. | |
15.
Work out the following referring back to the list you made of perfect squares
and cubes:
| |
Check your results in the following window. |
Numbers raised to fractional indices. | |
Thinking back to the previous example we can
safely say that:
as, referring to the rule to raise one
power to another power:
(81/3)3
= 81/3 * 3 = 81 = 8
In general we can say:
as:
(a1/n)n
= a1/n * n = a1 = a
Similarly, we can say:
| |
16. Express
the following in root form. Work out the value of the power. Use the following
window to check your answers. 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 indices. | |
Numbers raised to
fractional indices have the same properties as those raised to the power of a
whole number. Let's go over them again one by one:
The product of 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 is true for any base number or index, whether the
number be positive, negative, a whole number or a fraction. | |
17. Work out the answers to the following
products in your notebook and write them in index form:
a) 23/5 * 27/2
| |
Check your results in the following window. | |
Dividing powers with the same base number.
Similarly to the product rule, the
following general rule applies to both positive and negative indices:
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
| |
18. Work out the following divisions in
your notebook and write them in index form:
a) 27/3 : 24/3
| |
Check your results in the following window. | |
A product raised to a power.
A product raised to a power is equal to the product of the base numbers raised
to the same power.
(a*b)m
= am * bm | |
19. Express the following in product form:
a) (2*5)1/6
| |
Check your answers in the following window. | |
Dividing numbers raised to a power.
This works in a similar way to the product rule.
A division raised to a power is equal to one number raised to a power divided by
another number raised to the same power.
(a/b)m
= am / bm | |
20. Express the following in division form:
a) (18/2)5/6
| |
Check your results in the following window. | |
A power raised to a 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 | |
21. Express the
following in your notebook as numbers raised to just one power:
a) (21/3)7
| |
Check your results in the following window. | |
22. Use the rules in the examples above to help you to write a list of all the rules of roots and their operations. |
Fernando Arias Fernández-Pérez | ||
Spanish Ministry of Education. Year 2001 | ||
Except where otherwise noted, this work is licensed under a Creative Common License