## Note on Roots and Surds

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### 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: $$\sqrt{3} \times \sqrt[3]{4} \\ = \sqrt [2 \times 3]{3^3} \times \sqrt [3 \times 2]{4^2} \\ = \sqrt[6]{27} \times \sqrt [6] {16}\\ = \sqrt [6]{27 \times 16} \\ = \sqrt[6] {432}$$

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

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

For example: $$\sqrt2$$, $$\sqrt[3]{2}$$, $$\sqrt [4]{4}$$, $$\sqrt [5] {5}$$

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

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### Very Short Questions

Solution:

\begin{align*} &= 3\sqrt{2} + \sqrt [4] {2500} + \sqrt [4] {64} +6 \sqrt{8}\\&= 3\sqrt{2}+ \sqrt[4]{5 \times 5 \times 5 \times 5 \times 2 \times 2}+ \sqrt [4]{2 \times 2 \times 2 \times 2\times 2 \times 2}+ 6\sqrt{2 \times 2 \times 2}\\&= 3\sqrt{2}+5\sqrt[4]{4}+2\sqrt[4]{4}+12\sqrt{2}\\&=15\sqrt{2}+7\sqrt[4]{4} :_\text {Ans.} \end{align*}

\begin{align*} \sqrt [3] {16} + \sqrt [3] {54} - \sqrt [3] {250} &= \sqrt [3] {2^3 \times 2} + \sqrt [3] {3^3 \times 2} - \sqrt [3] {5^3 \times 2}\\ &= 2\sqrt [3]{2} + 3\sqrt [3]2 - 5\sqrt [3]2\\ &= 5\sqrt [3]2 - 5\sqrt [3]3\\ &= 0_{Ans}\end{align*}

\begin{align*} \frac {\sqrt {a + b} - \sqrt {a - b}}{\sqrt {a + b} + \sqrt {a - b}} &= \frac {\sqrt {a + b} - \sqrt {a - b}}{\sqrt {a + b} + \sqrt {a - b}} \times \frac {\sqrt {a + b} - \sqrt {a - b}}{\sqrt {a + b} - \sqrt {a - b}}\\ &= \frac {(\sqrt {a + b} - \sqrt {a - b})^2}{(\sqrt {a + b})^2 - (\sqrt {a - b})^2}\\ &= \frac {a + b + a - b - 2 \sqrt {a + b}\sqrt {a - b}}{a + b - a + b}\\ &= \frac {2a - 2\sqrt {a^2 - b^2}}{2b}\\ &= \frac {2(a - \sqrt {a^2 - b^2})}{2b}\\ &= \frac {a - \sqrt {a^2 - b^2}}b_{Ans}\\ \end{align*}

$$\frac {\sqrt y + \sqrt 5}{\sqrt y - \sqrt 5}$$ = 3

or, $$\sqrt y$$ + $$\sqrt 5$$ = 3$$\sqrt y$$ - 3$$\sqrt 5$$

or, 3$$\sqrt y$$ - $$\sqrt y$$ = $$\sqrt 5$$ + 3$$\sqrt 5$$

or, 2$$\sqrt y$$ = 4$$\sqrt 5$$

or, $$\sqrt y$$ = $$\frac {4\sqrt 5}{2}$$

or, $$\sqrt y$$ = 2$$\sqrt 5$$

Squaring on both sides,

($$\sqrt y$$)2= (2$$\sqrt 5$$)2

or, y = 4× 5

∴ y = 20Ans

$$\frac {2y + 3}{\sqrt y - 1}$$ = $$\frac 13$$ ($$\sqrt y$$ + 1)

or, 6y + 9 = ($$\sqrt y$$ + 1) ($$\sqrt y$$ - 1)

or, 6y + 9 = ($$\sqrt y$$)2 - (1)2

or, 6y + 9 = y - 1

or, 6y - y = -1 -9

or, 5y = - 10

or, y = $$\frac {-10}5$$

∴ y = -2Ans

$$\sqrt x$$ + 1 = 5 - $$\frac {\sqrt x -1}2$$

or, $$\sqrt x$$ + 1 - 5 = $$\frac {- (\sqrt x - 1)}{2}$$

or, $$\sqrt x$$ - 4 = $$\frac {-\sqrt x + 1}{2}$$

or, 2$$\sqrt x$$ - 8 + $$\sqrt x$$ = 1

or, 3$$\sqrt x$$ = 1 + 8

or, 3$$\sqrt x$$ = 9

or, $$\sqrt x$$ = $$\frac 93$$

or, $$\sqrt x$$ = 3

Squaring on both sides,

($$\sqrt x$$)2 = 32

∴ x = 9 Ans

$$\sqrt x$$ + $$\sqrt {x - 20}$$ = 10

or, $$\sqrt {x - 20}$$ = 10 - $$\sqrt x$$

Squaring on both sides,

($$\sqrt {x - 20}$$)2 = (10 - $$\sqrt x$$)2

or, x - 20 = 100 - 20$$\sqrt x$$ + x

or, 20$$\sqrt x$$ = 100 + 20

or, $$\sqrt x$$ = $$\frac {120}{20}$$

or, $$\sqrt x$$ = 6

Squaring on both sides,

($$\sqrt x$$)2 = 62

∴ x = 36Ans

$$\sqrt {4 (x + 1)}$$ = $$\sqrt {4x}$$ + 1

Squaring on both sides,

($$\sqrt {4 (x + 1)}$$)2 = ($$\sqrt {4x}$$ + 1)2

or, 4(x + 1) = 4x + 2$$\sqrt {4x}$$ + 1

or, 4x + 4 - 4x - 1 = 2$$\sqrt {4x}$$

or, $$\frac 32$$ = $$\sqrt {4x}$$

Again,

Squaring on both sides,

($$\sqrt {4x}$$)2 = ($$\frac 32$$)2

or, 4x = $$\frac 94$$

or, x = $$\frac 9{4 \times 4}$$

∴ x = $$\frac 9{16}_{Ans}$$

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