### Theory:

A radical notation is an expression used to denote a square root or $$n^{th}$$ root of a number or a variable.
Let $$a > 0$$ be a real number and '$$n$$' be a positive integer. Then $$\sqrt[n]{a} = b$$, if $$b^n = a$$ and $$b > 0$$.

The symbol $\sqrt[n]{}$ is called a radical, $$n$$ is the index of the radical and $$a$$ is called the radicand.

Let us find the root when $$n$$ is odd.

When $$n = 3$$ and $$a = 125$$, we have $$b^3 = 125$$ $$\Rightarrow b = \sqrt{125}$$ $$\Rightarrow b = 5$$

Therefore, when $$n$$ is odd, there is only one real root.

Consider finding the root, when $$n$$ is even.

When $$n = 2$$ and $$a = 81$$, we have $$b^2 = 81$$ $$\Rightarrow b = \sqrt{81}$$

In this case, we arrive at $$2$$ conclusions, $$9 \times 9 = 81$$ and $$(-9) \times (-9) = 81$$. And, we can say $$9$$ and $$-9$$ are roots of $$b$$.

But it is incorrect to write that $$\sqrt{81} = \pm{9}$$ because when $$n$$ is even, the roots are positive only if the radical symbol is positive. That is, $$\sqrt[n]{a}$$. Similarly, the roots are negative if the radical symbol is denoted by $$- \sqrt[n]{a}$$.

Therefore, we need to write $$\sqrt{81} = 9$$ and $$- \sqrt{81} = -9$$.
Fractional Index
Consider the radical notation $$\sqrt[n]{a} = b$$ and $$\sqrt{125} = 5$$

In this example, the root index tell us that the number of times the number($$5$$) multiplies itself to gives the radicand($$125$$).

Now, we are going to learn one more way of representing the powers and roots which involves the use of fractional indices.

In fractional index, we can write $$\sqrt[n]{a} = b$$ as $$a = b^{\frac{1}{n}}$$
Example:
Express the numbers the following numbers in the form $$4^{n}$$.

1. $$16$$

2. $$\frac{1}{64}$$

3. $$\sqrt{28}$$

4. $$256$$

Solution:

1. $$16 = 4 \times 4$$ $$= 4^2$$

2. $$\frac{1}{64} = \frac{1}{4 \times 4 \times 4}$$ $$= \frac{1}{4^{3}}$$ $$= 4^{-3}$$

3. $$\sqrt{20} = \sqrt{4} \times \sqrt{4} \times \sqrt{4} \sqrt{4} \times \sqrt{4} = (\sqrt{4})^5$$ $$= \left(4^{\frac{1}{2}}\right)^5$$ $$= 4^{\frac{5}{2}}$$

4. $$256 = 4 \times 4 \times 4 \times 4$$ $$= 4^4$$
Meaning of $$x^{\frac{m}{n}}$$, (where $$m$$ and $$n$$ are positive integers)
In the above example ($$3$$), we can see that the result is $$4^{\frac{5}{2}}$$. We say that this is of the form $$x^{\frac{m}{n}}$$ which is either $$n^{th}$$ root of the $$m^{th}$$ power of $$x$$ or $$m^{th}$$ power of the $$n^{th}$$ root of $$x$$.

That is, $$x^{\frac{m}{n}} = (x^m)^{\frac{1}{n}}$$ or $$(x^n)^{\frac{1}{m}} = \sqrt[n]{x^m}$$ or $$(\sqrt[n]{x})^m$$
Example:
Find the value of $$8^{\frac{5}{3}}$$ and $$256^{\frac{3}{4}}$$

Solution:

1. $$8^{\frac{7}{3}} = (\sqrt{8})^7 = (\sqrt{2^3})^7 = 2^7 = 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 = 128$$

2. $$256^{\frac{3}{4}} = (\sqrt{256})^3 = (\sqrt{4^4})^3 = 4^3 = 4 \times 4 \times 4 = 64$$