Roots and Surds

Roots

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\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

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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

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  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.

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 = \(\frac 9{16}_{Ans}\)

\begin{align*} \frac {3\sqrt 2}{\sqrt 6 - \sqrt 3} - \frac {4\sqrt 3}{\sqrt 6 - \sqrt 2} + \frac {2\sqrt 3}{\sqrt 6 + \sqrt 2} &= \frac {3\sqrt 2}{\sqrt 6 - \sqrt 3} \times \frac {\sqrt 6 + \sqrt 3}{\sqrt 6 + \sqrt 3} - \frac {4\sqrt 3}{\sqrt 6 - \sqrt 2} \times \frac {\sqrt 6 + \sqrt 2}{\sqrt 6 + \sqrt 2} + \frac {2\sqrt 3}{\sqrt 6 + \sqrt 2} \times \frac {\sqrt 6 - \sqrt 2}{\sqrt 6 - \sqrt 2}\\ &= \frac {3\sqrt {12} + 3\sqrt 6}{(\sqrt 6)^2 - (\sqrt 3)^2} - \frac {4\sqrt {18} + 4\sqrt {6}}{(\sqrt 6)^2 - (\sqrt 2)^2} + \frac {2\sqrt {18} - 2\sqrt 6}{(\sqrt 6)^2 - (\sqrt 2)^2}\\ &= \frac {3 \times 2\sqrt 3 + 3\sqrt 6}{6 - 3} - \frac {4 \times 3\sqrt 2 + 4\sqrt 6}{6 - 2} + \frac {2 \times 3\sqrt 2 - 2\sqrt 6}{6 - 2}\\ &= \frac {3(2\sqrt 3 + \sqrt 6)}{3} - \frac {4(3\sqrt 2 + \sqrt 6)}{4} + \frac {2(3\sqrt 2 - \sqrt 6)}{4}\\ &= 2\sqrt 3 + \sqrt 6 - 3\sqrt 2 - \sqrt 6 + \frac {3\sqrt 2 - \sqrt 6}2\\ &= \frac {2\sqrt 3 - 3\sqrt 2}{1} + \frac {3\sqrt 2 - \sqrt 6}{2}\\ &= \frac {4\sqrt 3 - 6\sqrt 2 + 3\sqrt 2 - \sqrt 6}{2}\\ &= \frac {4\sqrt 3 - 3\sqrt 2 - \sqrt 6}2_{Ans} \end{align*}

\begin{align*} \frac {1 + \sqrt 3}{1 - \sqrt 3} + \frac {2 + \sqrt 3}{2 - \sqrt 3} - \frac {1 + 2\sqrt 3}{2 + \sqrt 3} &= \frac {1 + \sqrt 3}{1 - \sqrt 3} \times \frac {1 + \sqrt 3}{1 + \sqrt 3}+ \frac {2 + \sqrt 3}{2 - \sqrt 3} \times \frac {2 + \sqrt 3}{2 + \sqrt 3} - \frac {1 + 2\sqrt 3}{2 + \sqrt 3} \times \frac {2 - \sqrt 3}{2 - \sqrt 3}\\ &= \frac {(1 + \sqrt 3)^2}{(1)^2 - (\sqrt 3)^2} + \frac {(2 + \sqrt 3)^2}{(2)^2 - (\sqrt 3)^2} - \frac {(2 - \sqrt 3 + 4\sqrt 3 - 2(\sqrt 3)^2)}{(2)^2 - (\sqrt 3)^2}\\ &= \frac {1^2 + 2\sqrt 3 +(\sqrt 3)^2}{1 - 3} + \frac {2^2 + 4\sqrt 3 + (\sqrt 3)^2}{4 - 3} - \frac {(2 + 3\sqrt 3 - 6)}{4 - 3}\\ &= \frac {1 + 3 + 2\sqrt 3}{-2} + \frac {4 + 4\sqrt 3 + 3}{1} - \frac {(3\sqrt 3 - 4)}{1}\\ &= \frac {4 + 2\sqrt 3}{-2} + 7 + 4\sqrt 3 - 3\sqrt 3 + 4\\ &= \frac {2(2 + \sqrt 3)}{-2} + 11 + \sqrt 3\\ &= -2 -\sqrt 3 + 11 + \sqrt 3\\ &= 9_{Ans}\\ \end{align*}

\(\frac {3\sqrt x - 4}{\sqrt x + 2}\) = \(\frac {15 + 3\sqrt x}{\sqrt x + 40}\)

or, (3\(\sqrt x\) - 4) (\(\sqrt x\) + 40) = (15 + 3\(\sqrt x\)) (\(\sqrt x\) + 2)

or, 3x + 120\(\sqrt x\) - 4\(\sqrt x\) - 160 = 15\(\sqrt x\) + 30 + 3x + 6\(\sqrt x\)

or, 3x - 3x + 116\(\sqrt x\) - 160 = 21\(\sqrt x\) + 30

or, 116\(\sqrt x\) - 21\(\sqrt x\) = 160 + 30

or, 95\(\sqrt x\) = 190

or, \(\sqrt x\) = \(\frac {190}{95}\)

or, \(\sqrt x\) = 2

Squaring on both sides,

(\(\sqrt x\))2 = (2)2

∴ x = 4Ans

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

or, \(\frac {x - 1}{\sqrt x + 1}\) = \(\frac {8 + \sqrt x - 1}2\)

or,\(\frac {x - 1}{\sqrt x + 1}\) = \(\frac {7 + \sqrt x}2\)

or, 2x - 2 = 7\(\sqrt x\) + 7 + x + \(\sqrt x\)

or, 2x - x - 2 = 8\(\sqrt x\) + 7

or, x - 2 - 7 = 8\(\sqrt x\)

or, x - 9 = 8\(\sqrt x\)

Squaring on both sides,

(x - 9)2 = (8\(\sqrt x\))2

or, x2 - 18x + 81 = 64x

or, x2 - 18x + 81 - 64x = 0

or, x2 - 82x + 81 = 0

or, x2 - 81x - x + 81 = 0

or, x(x - 81) - 1 (x - 81) = 0

or, (x - 81) (x - 1) = 0

Either,

x - 81 = 0

∴ x = 81

Or,

x - 1 = 0

∴ x = 1

Putting the value of x = 1 is false.

∴ x = 81Ans

\(\frac {\sqrt {x + 4} + \sqrt {x - 4}}{\sqrt {x + 4} - \sqrt {x - 4}}\) = 3

or, \(\sqrt {x + 4}\) + \(\sqrt {x - 4}\) = 3\(\sqrt {x + 4}\) - 3\(\sqrt {x - 4}\)

or, \(\sqrt {x - 4}\) + 3\(\sqrt {x - 4}\) = 3\(\sqrt {x + 4}\) - \(\sqrt {x + 4}\)

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

or, \(\frac {\sqrt {x + 4}}{\sqrt {x - 4}}\) = \(\frac 42\)

or, \(\frac {\sqrt {x + 4}}{\sqrt {x - 4}}\) = 2

Squaring on both sides,

( \(\frac {\sqrt {x + 4}}{\sqrt {x - 4}}\))2 = 22

or, \(\frac {x + 4}{x - 4}\) = 4

or, x + 4 = 4x - 16

or, 4x - x = 16 + 4

or, 3x = 20

∴ x = \(\frac {20}3_{Ans}\)

\(\sqrt x\) + \(\sqrt {x + 13}\) = \(\frac {91}{\sqrt {x + 13}}\)

or, \(\sqrt {x + 13}\) (\(\sqrt x\) + \(\sqrt {x + 13}\)) = 91

or, \(\sqrt x\) \(\sqrt {x + 13}\) + (\(\sqrt {x + 13}\))2 = 91

or, \(\sqrt x\) \(\sqrt {x + 13}\) + x + 13 = 91

or, \(\sqrt x\) \(\sqrt {x + 13}\) = 91 - 13 - x

or, \(\sqrt x\) \(\sqrt {x + 13}\) = 78 - x

Squaring on both sides,

(\(\sqrt x\) \(\sqrt {x + 13}\))2 = (78 - x)2

or, x (x + 13) = (78)2 - 2 . 78 . x + x2

or, x2 + 13x = 6084 - 156x + x2

or, x2 + 13x + 156x - x2 = 6084

or, 169x = 6084

or, x = \(\frac {6084}{169}\)

∴ x = 36Ans

\(\sqrt {4x + 5}\) - \(\sqrt x\) = \(\sqrt {x + 3}\)

Squaring on both sides,

(\(\sqrt {4x + 5}\) - \(\sqrt x\))2 = (\(\sqrt {x + 3}\))

or, (\(\sqrt {4x + 5}\))2 - 2\(\sqrt {4x + 5}\) . \(\sqrt x\) + (\(\sqrt x\))2 = x + 3

or, 4x + 5 - 2\(\sqrt {4x^2 - 5x}\) + x - x - 3 = 0

or, 4x + 2 =2\(\sqrt {4x^2 - 5x}\)

Again,

Squaring on both sides,

(4x + 2)2 = (2\(\sqrt {4x^2 - 5x}\))2

or, 16x2 + 16x + 4 = 4(4x2 + 5x)

or, 16x2 + 16x + 4 = 16x2 + 20x

or, 16x2 + 16x + 4 - 16x2 - 20x = 0

or, -4x + 4 = 0

or, -4x = -4

or, x = \(\frac 44\)

∴ x = 1Ans

\(\sqrt {x - \sqrt {1 - x}}\) = 1 - \(\sqrt x\)

Squaring on both sides,

(\(\sqrt {x - \sqrt {1 - x}}\))2 = (1 - \(\sqrt x\))2

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

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

or, x - x - \(\sqrt {1 - x}\) = 1 - 2\(\sqrt x\)

or, -\(\sqrt {1 - x}\) = 1 - 2\(\sqrt x\)

Again,

Squaring on both sides,

(-\(\sqrt {1 - x}\))2 = (1 - 2\(\sqrt x\))2

or, 1 - x = 1 - 4\(\sqrt x\) + 4x

or, 1 - x - 1 - 4x = -4\(\sqrt x\)

or, -5x = -4\(\sqrt x\)

Again,

Squaring on both sides,

(-5x)2 = (-4\(\sqrt x\))2

or, 25x2 = 16x

or, 25x2 - 16x = 0

or, x(25x - 16) = 0

Either,

x = 0 (impossible)

Or,

25x - 16 = 0

or, 25x = 16

∴ x = \(\frac {16}{25}\)

∴ x = \(\frac {16}{25}_{Ans}\)

\(\sqrt {x^2 - 3x + 3}\) + \(\sqrt {x^2 - x + 1}\) = 2

or,\(\sqrt {x^2 - 3x + 3}\) = 2 -\(\sqrt {x^2 - x + 1}\)

Squaring on both sides,

(\(\sqrt {x^2 - 3x + 3}\))2 = (2 -\(\sqrt {x^2 - x + 1}\))2

or, x2 - 3x + 3 = 4 - 4\(\sqrt {x^2 - x + 1}\) + (\(\sqrt {x^2 - x + 1}\))2

or, x2 - 3x + 3 - 4 = - 4\(\sqrt {x^2 - x + 1}\) + x2 - x + 1

or, x2 - 3x - 1 - x2 + x - 1 =- 4\(\sqrt {x^2 - x + 1}\)

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

or, -2 (x + 1) =- 2\(\sqrt {x^2 - x + 1}\)

or, x + 1 = \(\sqrt {x^2 - x + 1}\)

Again,

Squaring on both sides,

(x + 1)2 = (\(\sqrt {x^2 - x + 1}\))2

or, x2 + 2x + 1 = 4(x2 - x + 1)

or, x2 + 2x + 1 = 4x2 - 4x + 4

or, 4x2 - x2 - 4x - 2x + 4 - 1 = 0

or, 3x2 - 6x + 3 = 0

or, 3(x2- 2x + 1) = 0

or, x2 - 2x + 1 = 0

or, (x)2 - 2 . x . 1 + (1)2 = 0

or, (x - 1)2 = 0

Removing square from both sides,

x - 1 = 0

∴ x = 1Ans

\(\frac {a - 2}{a^2 - 2a + 4}\) + \(\frac {a + 2}{a^2 + 2a + 4}\) - \(\frac {16}{a^4 + 16 + 4a^2}\)

= \(\frac {(a - 2) (a^2 + 2a + 4) + (a + 2) (a^2 - 2a + 4)}{(a^2 - 2a + 4) (a^2 + 2a + 4)}\) - \(\frac {16}{(a^2)^2 + 2 . a^2 . 4 + (4)^2 - 4a^2}\)

= \(\frac {a^3 - 2^3 + a^3 + 2^3}{(a^2 - 2a + 4) (a^2 + 2a + 4)}\) - \(\frac {16}{(a^2 + 4)^2 - (2a)^2}\)

= \(\frac {2a^3}{(a^2 - 2a + 4) (a^2 + 2a + 4)}\) - \(\frac {16}{(a^2 + 2a + 4) (a^2 - 2a + 4)}\)

= \(\frac {2a^3 - 16}{(a^2 - 2a + 4) (a^2 + 2a + 4)}\)

= \(\frac {2(a^3 - 8)}{(a^2 - 2a + 4) (a^2 + 2a + 4)}\)

= \(\frac {2(a^3 - 2^3)}{(a^2 - 2a + 4) (a^2 + 2a + 4)}\)

= \(\frac {2(a - 2) (a^2 + 2a + 4)}{(a^2 - 2a + 4) (a^2 + 2a + 4)}\)

= \(\frac {2(a - 2)}{(a^2 - 2a + 4)}_{Ans}\)

0%
  • √27 + √75 - 8√3

    five


    zero


    six


    eight


  • √32 + √8 - √72

    two


    one


    five


    zero


  • √12 - √75 + √48

    √8


    √2


    √7


    √3


  • √50 + √18 - 8√2

    zero


    five


    three


    nine


  • √125 - √45 + √5

    3√6


    3√9


     3√2


    3√5


  • (sqrt {3x+13}=5)

    2


    8


    5


    4


  • (sqrt {2x+1}-3=0)

    9


    2


    7


    4


  • (sqrt {x+3}-1=2)

    10


    8


    9


    6


  • (sqrt {x+2}=3)

    9


    5


    7


    2


  • (sqrt {2a-1}=3)

    5


    2


    9


    12


  • 4(sqrt {a-3}=sqrt{5a+7})

    3
    9
    5
    2
  • (frac{x-1}{sqrt{x}-1}=2+frac{sqrt x-5}{3})

    1
    5
    6
    7
  • (frac{x-6}{sqrt{x}-1}=4-frac{sqrt x-2}{3})

    18
    12
    16
    20
  • (frac{3x-1}{sqrt{3x+6}}=1+frac{sqrt 3x-1}{2})

    1
    6
    3
    2
  • (sqrt{x+3} ={sqrt{2x+4}}-1)

    9
    6
    2
    7
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