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Long Questions :

10. Simplify:

(a) 5/125-645 + 2√5

(b) (d) 20 + 4sqrt(80 - 2sqrt(180); 40) + 3 * root(625, 3) - root(320, 3)

(c) 4√90-6√160 + √245 - 5/125

11. Arrange the following surds in ascending order :

(a) 5, √2 and √7

(b) √7, 15 and √3

(c) √5, 11 and 2/3

12. Rationalize the denominators and simplify :

(a) (7 + sqrt(3))/(7 - sqrt(3))

b (2 + sqrt(5))/(2 - sqrt(5))

(c) (sqrt(5) + sqrt(2))/(sqrt(5) - sqrt(2))
383×2=6

3.6

Rationalization of Surds

When the product of two surds is a rational number, each of them is said to be the rationalizing factor of the other.

Example: √3+√2 is rationalizing factor of √3-√2. Then, (√3+√2) x (√3-√2)=3-2 = 1.

The process from which a surd is changed to a rational number by multiplying it with a suitable factor is called the rationalization of the surd..

The following are some basic steps for rationalization.

(i)

Multiply the given surd by a simplest rationalizing factor.

Example: 4√2 × √2 = 4 × 2 = 8

Here √2 is a simplest rationalizing factor.

(ii) Multiply a binomial surd by its conjugate.

Example: √3+√2, its conjugate is √3-√2

Now, (√3 + √2) x (√3-√2) = (√3)² - (√2)² = 3-2 = 1

Here, √3-√2 is a rationalizing factor of √3 + √2.

(iii) When a surd is in the form of quotient, multiply both the numerator and denominator by conjugate of the denominator to make denominator a rational number.

2+√5 Example: 5-2

44

(multiplying numerator and denominator by √5 + 2)

2+55+2 √5-25 5+2 = = (√5+2)²(5)+2.5.2+2 (5)-2 5-4 5+ 4/5 + 4 = 9+4/5 1
n is

Example:

(i) √7+ √5 and √7-√5 are conjugate surds.

(ii) 2√3-2√2 and 2√3 2sqrt(2) are conjugate surds.

Laws of Surd

Let a and b be two positive real numbers and m, n and p be integers. Then.

he surd.

er of the

of unlike

(i) a = a

(ii) a. V = Vab

na

(iv) sqrt[n] ( sqrt[n] a = root(a, mn) = root(root(a, m), n)

(v) mn

3.5

Operation of Surds

e.g. 3 = 3

e.g. sqrt(5) * sqrt(3) = sqrt(5 * 3) = sqrt(15)

e.g. (root(4, 3))/(root(2, 3)) = root(4/2, 3) = root(2, 3)

e.g. == 1/4

e.g. root(2, 3) = root(2 ^ 4, 3 * 4) = root(16, 12) = 2 ^ (4 * (- 1/12)) = 2 ^ (1/3)

(i) Addition and Subtraction of Surds

Two or more like surds can be added or subtracted. To add or subtract surds, follow the given ways:

(a) Express all the like surds into simplest form.

(b) Add or subtract the coefficients of like terms keeping the irrational factor same.

Examples:

(i) overline 7 + 3sqrt(7) = (1 + 3) * sqrt(7) = 4sqrt(7)

(ii) 4√3-2√3()

(iii) √18 + √50

= √9x2+√25 × 2 = 3sqrt(2) + 5sqrt(2) = (3 + 5) * sqrt(2) = 8sqrt(2)

(iv) overline 2 + sqrt(8) - sqrt(18)

= 4√2 + 2√2-3√2

=(4+2-3)√2 = 3√2

ed surd.

(ii) Multiplication and Division of Surds:

Two or more surds of same order c
an be multiplied.

Example:

Ivedantal Erru(11) 5x25 =45×25-5-5

(iii) sqrt(7) * 4sqrt(7) sqrt * 7

Wor

12x7-84

Note: Va Vab

Beaample 17

Add: 4/7+8/7

Solution:

Here. 4/7+/7

A surd can be divided by another surd of the same order.

Example:

(1) 1sqrt(7) / 2 * sqrt(7) 4 sqrt prime 2sqrt(7) = 2 =

( t + mv

Example 2.

Subtract: 8/5-

Solution:

Here, 8/5-5/5

(H) 3/72+V6

3/72 72 3V9

(8-55

3/8=3×2=6

sample 3:

Multiplys

3.6

Rationalization of Surds

(i) sqrt(3) * sqrt(2) = sqrt(3 * 2) = sqrt(6)

al Mathematics Book 8

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