Lesson Introduction to Radicals

Source code of 'Introduction to Radicals'

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About Shruti_Mishra: I am a maths graduate from India and am currently persuing masters in Operations Research.

<a href="Nth_root.wikipedia">Radicals</a> are <a href="exponent.wikipedia">exponents</a> with fractional powers. Alternatively it is inverse of power. For example, {{{(5)^(1/3)}}} is 5 with fractional power of 1/3. Alternatively this means that the if x = {{{(5)^(1/3)}}}, then {{{5 = x^3}}}. This forms the first property of a <a href="Nth_root.wikipedia">radical<a/> which we would see in the chapter - <a href="prop_radicals.lesson"> Properties of Radicals</a>. A <a href="Nth_root.wikipedia">radical</a> is represented using a {{{sqrt(x)}}} sign and {{{root(n,x)}}} means {{{(x)^(1/n)}}}. Radicals are called as roots. Thus {{{root(n,x)}}} 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 {{{y = root(n,x)}}}, then {{{ x = y^n}}}.

The <a href="Nth_root.wikipedia">radicals</a> which cannot be simplified into integers are <a href="irrational_number.wikipedia">irrational</a>. When the <a href="Nth_root.wikipedia">radical</a> is left completely within the <a href="Nth_root.wikipedia">radical</a> sign and not simplified, it is also known as surd. Let us consider a few examples of <a href="Nth_root.wikipedia">radicals</a> to understand its usage. We would start with square roots.

Square roots
We know that {{{2^2 = 4}}} and {{{3^2 = 9}}}. Thus we can inverse these and get the <a href="Nth_root.wikipedia">radicals</a> as {{{sqrt(4)}}} and {{{sqrt(9)}}}. Remember that square roots do not need '2' to be put over the <a href="Nth_root.wikipedia">radical</a> sign. In these examples, the numbers in the radical are perfect squares and would yield integral answers. However, consider a radical like {{{sqrt(2)}}}, 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 {{{sqrt(3)}}} the value would be close to 1.73205080756.

Cube roots
We know that {{{2^3 = 8}}} and {{{3^3 = 27}}}. Thus we can inverse these and get the <a href="Nth_root.wikipedia">radicals</a> as {{{root(3,8)}}} and {{{root(3,27)}}}. In these examples, the numbers in the radical are perfect cubes and would yield integral answers. However, consider a radical like {{{root(3,5)}}}, 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 <a href="exponent.wikipedia">exponents</a> such as {{{2^4 = 16}}} and {{{3^5 = 243}}}. Thus we can inverse these and get the <a href="Nth_root.wikipedia">radicals</a> as {{{root(4,16)}}} and {{{root(5,243)}}}. In these examples, the numbers in the radical are perfect powers and would yield integral answers. However, consider a radical like {{{root(5,28)}}}, 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 <a href="Nth_root.wikipedia">radicals</a> are (approximate):
{{{sqrt(2) = 1.4142}}}
{{{sqrt(3) = 1.7320}}}
{{{sqrt(5) = 2.2360}}}
{{{root(3,2) = 1.2599}}}
{{{root(3,3) = 1.4422}}}
Remember that and radical with 1 would always have value 1, i.e. {{{root(n,1) = 1}}} for all n.

Other examples are {{{root(5,17)}}}, {{{root(7,21)}}}, {{{root(6,2)}}} etc.