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**1. Rationalize the denominator of each of the following:**

**(i) ****(ii) (iii) (iv) (v) (vi) (vii) **

**Solution: (i) ** For rationalizing the denominator, multiply both numerator and denominator with

**(ii) **For rationalizing the denominator, multiply both numerator and denominator with

**(iii) **For rationalizing the denominator, multiply both numerator and denominator with

**(iv) **For rationalizing the denominator, multiply both numerator and denominator with

**(v) **For rationalizing the denominator, multiply both numerator and denominator with

**(vi) **For rationalizing the denominator, multiply both numerator and denominator with

**(vii) **For rationalizing the denominator, multiply both numerator and denominator with

**2. Find the value to three places of decimals of each of the following.** **It is given that ** ** = 1.414,** **= 1.732,** ** = 2.236, ** **= 3.162.**

**(i) ****(ii) ****(iii) ****(iv) ****(v) ****(vi) **

**Solution:** Given, ** = 1.414,** **= 1.732,** ** = 2.236, ** **= 3.162.**

**(i) ** Rationalizing the denominator by multiplying both numerator and denominator with

**(ii) ** Rationalizing the denominator by multiplying both numerator and denominator with

**(iii) ** Rationalizing the denominator by multiplying both numerator and denominator with

**(iv) ** Rationalizing the denominator by multiplying both numerator and denominator with

**(vi) ** Rationalizing the denominator by multiplying both numerator and denominator with

**3. Express each one of the following with rational denominator:**

**(i) ****(ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) **

**Solution: (i) ** Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a^{2}âˆ’b^{2})

**(ii) **Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a^{2} âˆ’ b^{2})

**(iii) **Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a^{2}âˆ’b^{2})

**(iv) **Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a^{2}âˆ’b^{2}) =

**(v) **Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a^{2}âˆ’b^{2}) =

**(vi) **Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a^{2}âˆ’b^{2}) =

**(vii) **Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a^{2}âˆ’b^{2}) As we know , (a - b)^{2} = (a^{2}âˆ’2Ã—aÃ—b+b^{2})

**(viii) **Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a^{2}âˆ’b^{2}) =

**(ix) ** Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a^{2}âˆ’b^{2})

**4. Rationalize the denominator and simplify:**

**(i) **

**(ii) **

**(iii) **

**(iv) **

**(v) **

**(vi) **

**Solution: (i) **Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) As we know , (a - b) 2 = (a^{2}âˆ’2Ã—aÃ—b+b^{2})

**(ii) **Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a^{2}âˆ’b^{2}) =

**(iii) **Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a^{2}âˆ’b^{2})

**(iv) **Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a^{2}âˆ’b^{2}) =

**(v) **Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a^{2}âˆ’b^{2}) =

**(vi) **Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

**5. Simplify: **

**(i) **

**(ii) **

**(iii) **

**(iv) **

**(v) **

**Solution:** **(i) ** Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor and the rationalizing factor

Now, (a + b) (a - b) = (a^{2}âˆ’b^{2}) =

As we know , (a - b)^{2} = (a^{2}âˆ’2Ã—aÃ—b+b^{2}) =

**(ii) **Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor and the rationalizing factor =

Now as we know, (a + b) (a - b) = (a^{2}âˆ’b^{2}), (a - b)^{2} = (a^{2}âˆ’2Ã—aÃ—b+b^{2}) and (a+b)^{2} = (a^{2}+2Ã—aÃ—b+b^{2})

**(iii) **Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor and the rationalizing factor

Now as we know, (a + b) (^{2}âˆ’b^{2}) =

**(iv) **Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor the rationalizing

Since, (a + b) (^{2}âˆ’b^{2})

**(v) ** Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor the rationalizing factor and the rationalizing factor

Since, (a + b) (a - b) = (a^{2}âˆ’b^{2}) =

**6. In each of the following determine rational numbers a and b:**

**(i) **

**(ii) **

**(iii) **

**(iv) **

**(v) **

**(vi) **

**Solution:** (i) Given, ** **Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a^{2}âˆ’b^{2})

On comparing the rational and irrational parts of the above equation, we get, a = 2 and b = 1

**(ii) **Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (

On comparing the rational and irrational parts of the above equation, we get,

a = 3 and b = 2

**(iii) **Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (

On comparing the rational and irrational parts of the above equation, we get,

**(iv) **Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (

On comparing the rational and irrational parts of the above equation, we get,

a = -1 and

b = 1

**(v) **Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a^{2}âˆ’b^{2}) =

On comparing the rational and irrational parts of the above equation, we get,

**(vi) **Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (^{2}âˆ’b^{2})

On comparing the rational and irrational parts of the above equation, we get,

**7. If x = 2 + ** **find the value of **

**Solution: ** Given, x = 2 + To find the value of

We have, **x = 2 + **

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

Since, (a + b) (^{2}âˆ’b^{2})

We know that,

Putting the value of in the above equation, we get,

**8. If x = 3 + ** **find the value of **

**Solution: ** Given, x = 3 +

To find the value

Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

Since, (a + b) (a - b) = (a^{2}âˆ’b^{2})

**9. Find the value of **

**Solution.** Given, Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

Since, (a + b) (a - b) = (a^{2}âˆ’b^{2}) = = 3(2.236+1.732) = 3(3.968) = 11.904

**10. Find the values of each of the following correct to three places of decimals, it being given that ** **= 1.414,** **= 1.732,** **= 2.236,** **= 2.4495,** **= 3.162**

**Solution. (i) ** Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

Since, (a + b) (a - b) = (a^{2}âˆ’b^{2}) =

**(ii) ** Rationalizing the denominator by multiplying both numerator and denominator with the rationalizing factor

As we know, (a + b) (a - b) = (a^{2}âˆ’b^{2}) = 7 + 7.07 = 14.07

**11. If x = ** **find the value of 4x ^{3}+2x^{2}âˆ’8x+7.**

**Solution:** Given,

and given to find the value of 4x^{3}+2x^{2}âˆ’8x+7

Now, squaring on both the sides, we get, (2xâˆ’1)^{2} = 3

4x^{2}âˆ’4x+1 = 3

4x^{2}âˆ’4x+1âˆ’3 = 0

4x^{2}âˆ’4xâˆ’2 = 0

2x^{2}âˆ’2xâˆ’1 = 0

Now taking 4x^{3}+2x^{2}âˆ’8x+7

2x(2x^{2}âˆ’2xâˆ’1)+ 4x^{2}+2x+2x^{2}âˆ’8x+7

2x(2x^{2}âˆ’2xâˆ’1)+ 6x^{2}âˆ’6x+7

As, 2x^{2}âˆ’2xâˆ’1=0

2x(0)+3(2x^{2}âˆ’2xâˆ’1))+7+3

0+3(0)+10

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