Roots

\begin{align*} \text {We know that} \: 3 \times 3 &= 9 \\ or, \: 3^2 &= 9 \\ or, \: 3^{2 \times \frac {1}{2}} &= 9^{\frac{1}{2}} \\ or, \: 3 &=9^{\frac{1}{2}} \\ or, \: 3 &= \sqrt{9} \\ \end{align*}

The square root of 9 is 3.

\begin{align*} similarly, \: 5 \times 5\times 5 &= 125 \\ or, \: 5^3 &= 125 \\ or, 5^{3 \times \frac {1}{3}} &= (125)^{\frac{1}{3}} \\ or, \: 5 &= (125)^{\frac{1}{3}} \\ or, \: 5 &= \sqrt[3]{125} \end{align*}

The cube root of 125 is 5.
In case of equation, root indicates the value of variable.
e.g. x + 2 = 0, x = -2,
So, -2 is the root of x
e.g. x² - 4 = 0,
(x - 2)(x +2) = 0

Either, (x - 2) = 0 or, x = 2
or, (x + 2) = 0, x = -2

So, -2 and 2 are roots of x² -4 =0

Surds

Surds are numerical expressions containing an irrational number. Surds may be quadratic, bi-quadratic, cubic etc.

For example: \( \sqrt{2}, \sqrt[3]{3}, \sqrt [4]{4}, \sqrt [5] {5}\)

The surds cannot be written in the form of \( \frac {p}{q} \) q≠ 0, so they are irrational numbers.

Types of surds

1. Pure surds:
   If the natural number is completely inside the root or radical, the surd is called pure surd. \( \sqrt {2}, \sqrt [3]{5} \) are pure surds.
2. Mixed surd:
   If the integers are inside and outside the radical, the surds is called mixed surd. 2\( \sqrt [2] {5} \) is a mixed surd.
3. Simple surd:
   The single surd which may be pure or mixed is called simple surd. \( \sqrt [4]{4} \: and \:3 \sqrt [4]{6} \) are simple surds.
4. Compound surd
   The sum or difference of two pure or mixed surds is called compound surd. \( \sqrt {5} + 2 , \: \sqrt[3]{2} + 6 \) are compoud surds.
5. Like surds
   If the power (degree) of surds and number inside the root is same the surds are called like surds.\( \sqrt {5} , 2\sqrt {5} \) are like surds.
6. Unlike surds
   If a power of root is different or numbers inside the root are different the surds are called, unlike surds.\( \sqrt {5} , \sqrt [4]{4}\:and \sqrt {2}\) are unlike surds.

Four fundamental operations on surds

1. Additional and subtraction of surds:
   The addition and subtraction of like surds can exist, unlike surds are neither added nor subtracted. For example: \( 7 \sqrt{2} + 8 \sqrt{2} = (7 + 8) \sqrt{2} = 15 \sqrt{2} \)
2. Multiplication and division of surds:
   If the order of surds is same, we can put them within common root and perform the multiplication and division just like in arithmetic.
   For example: a. \(\sqrt {2} \times \sqrt{3} \\ = \sqrt {2 \times 3} \\ = \sqrt {6}\)   b. \( \sqrt{10}÷ \sqrt{2} \\ = \sqrt {\frac{10}{2}} \\ = \sqrt{}5 \)
   If the order of surds are not same then we shall reduce these surds in the same order.
   For example: \(\sqrt{3} \times \sqrt[3]{4} \\ = \sqrt [2 \times 3]{3^3} \times \sqrt [3 \times 2]{4^2} \\ = \sqrt[6]{27} \times \sqrt [6] {16}\\ = \sqrt [6]{27 \times 16} \\ = \sqrt[6] {432} \)

\(\boxed{\text {Note: Same process can be applied for division also}}\)