Radicals are
exponents with fractional powers. Alternatively it is inverse of power. For example,
is 5 with fractional power of 1/3. Alternatively this means that the if x =
, then
. This forms the first property of a
radical which we would see in the chapter -
Properties of Radicals. A
radical is represented using a
sign and
means
. Radicals are called as roots. Thus
is called as 'n'th root of x. There are special names for n = 2 and 3, which are square root and cube root respectively.
Property
If
, then
.
The
radicals which cannot be simplified into integers are
irrational. When the
radical is left completely within the
radical sign and not simplified, it is also known as
surd. Let us consider a few examples of
radicals to understand its usage. We would start with square roots.
Square roots
We know that
and
. Thus we can inverse these and get the
radicals as
and
. Remember that square roots do not need '2' to be put over the
radical sign. In these examples, the numbers in the radical are perfect squares and would yield integral answers. However, consider a radical like
, where 2 is not a perfect square. This radical could not be simplified into an integral answer. The value on solving for this would be 1.41421356237 with an infinite decimal sequence. Similarly, for
the value would be close to 1.73205080756.
Cube roots
We know that
and
. Thus we can inverse these and get the
radicals as
and
. In these examples, the numbers in the radical are perfect cubes and would yield integral answers. However, consider a radical like
, where 5 is not a perfect cube. This radical can not be simplified into an integral answer. The value on solving for this would be close to 1.709975946676 with an infinite decimal sequence.
Higher roots
For higher roots, we consider
exponents such as
and
. Thus we can inverse these and get the
radicals as
and
. In these examples, the numbers in the radical are perfect powers and would yield integral answers. However, consider a radical like
, where 2 is not a perfect square. This radical could not be simplified into an integral answer. The value on solving for this would be close to 1.9472943612303362 with an infinite decimal sequence.
Some other examples and values of common radicals
Some common values of
radicals are (approximate):
Remember that and radical with 1 would always have value 1, i.e.
for all n.
Other examples are
,
,
etc.
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