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Simplify 5/(4 – √3)

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  • Last Updated : 27 May, 2022

Rationalization is a method that is used in elementary mathematics to eliminate the irrational number in the denominator. To rationalize the denominator, numerous rationalizing strategies are applied. Rationalization is the act of eliminating a radical or an imaginary number from the denominator of an algebraic fraction. That is, in a fraction, eliminate the radicals such that the denominator only includes a rational number. Rationalizing is the process of multiplying one surd by another to get a rational number. The rationalization factor is the surd that is utilized to multiply.

To rationalize √a we need another √a: √a × √a = a.

To rationalize p + √q we need a rationalizing factor p -√q, 

(p +√q) × (p -√q) = (p)2 – (√q)2 = p2 – q.

The rationalizing factor of 5√3 is √3,

5√3 × √3 

= 5 × 3 

= 15

Rationalize the Denominator with Conjugate 

A conjugate is a surd that is similar but has a different sign. The conjugate of (6 + √2) is (6 – √2). The conjugate is the rationalizing factor in the process of rationalizing a denominator. The following is the method for rationalizing the denominator with its conjugate.

  • Step 1: Multiply the denominator and numerator by a conjugate that removes the radicals in the denominator.
  • Step 2: We must ensure that all of the surds in the specified fraction are simplified.
  • Step 3: If necessary, we can simplify the fraction even further.

Simplify 5/(4 – √3)

Solution: 

Given: 5/(4 – √3)

Multiply the denominator and numerator by a conjugate that removes the radicals in the denominator.

Therefore, {5/(4 – √3)} ×  {(4 + √3) / (4 + √3)}

= {5 (4 + √3)} / {(4 – √3)(4 + √3)}

= (20 + 5√3) / {(4)2 – (√3)2}

= (20 + 5√3) / {16 – 3}

= (20 + 5√3)/13

Similar Questions

Question 1: Rationalize the denominator and simplify, if possible, 6/(2 – √4)?

Solution: 

Given: 6/(2 – √4)

Multiply the denominator and numerator by a conjugate that removes the radicals in the denominator.

Therefore ,{ 6/(2 – √4)} × {(2 + √4) / (2 + √4)}

= {6 (2 + √4)} / {(2 – √4)(2 + √4)}

= (12 + 6√4) / { (2)2 – (√4)2}

= (12 + 6√4) / { 4 – 4 }

= (12 + 6√4)/0

= (12 + 6√4)

Question 2: Rationalize the denominator and simplify, if possible, 5/(5 + √4)?

Solution: 

Given: 5/(5 + √4)

Multiply the denominator and numerator by a conjugate that removes the radicals in the denominator.

Therefore, {5/(5 + √4)} × {(5 – √4) / (5 – √4)}

= {5 (5 – √4)} / {(5 + √4)(5 – √4)}

= (25 – 5√4) / {(5)2  – (√4)2}

= (25 – 5√4) / {25 – 4}

= (25 – 5√4)/21

Question 3: Rationalize the denominator and simplify, if possible, 7 /( √5 + √6)?

Solution: 

Given: 7/(√5 + √6)

Multiply the denominator and numerator by a conjugate that removes the radicals in the denominator.

Therefore, {7/(√5 + √6)} × {(√5 – √6) / (√5 – √6)}

= {7 (√5 – √6)} / {(√5 + √6)(√5 – √6)}

= (7√5 – 7√6) / {(√5)2  – (√6)2}

= (7√5 – 7√6) / {5 – 6}

= (7√5 – 7√6)/(-1)

= – {(7√5 – 7√6)}

= – 7√5 + 7√6 

Question 4: Rationalize the denominator and simplify, if possible, 2 /( √6 + 5)?

Solution: 

Given: 2 /( √6 + 5)

Multiply the denominator and numerator by a conjugate that removes the radicals in the denominator.

Therefore, {2/(√6 + 5)} × {(√6 – 5) / (√6 – 5)}

= {2 (√6 – 5)} / {(√6 + 5)(√6 – 5)}

= (2√6 – 10) / {(√6)2  – (5)2}

= (2√6 – 10) / {6 – 25}

= (2√6 – 10)/(-19)

= – (2√6 – 10)/19

Question 5: Rationalize the denominator and simplify, if possible, (2 +√6) /( 4 + √6)?

Solution: 

Given: (2 +√6) /( 4 + √6) 

Multiply the denominator and numerator by a conjugate that removes the radicals in the denominator.

Therefore, {(2 + √6)/(4 + √6 )} × {(4 – √6) / (4 – √6)}

= {(2 + √6) (4 – √6)} / {(4 + √6)(4 – √6)}

= (8 – 2√6 + 4√6 – 6) / {(4)2 – (√6 )2}

= (2 + 2√6) / {16 – 6}

= (2 + 2√6 )/(10)

= {2(1 + √6)}/10

= (1 + √6 )/5

Question 6: Rationalize the denominator and simplify, if possible, (3 +√8) /( 2 + √8)?

Solution: 

Given: (3 + √8) / (2 + √8)

Multiply the denominator and numerator by a conjugate that removes the radicals in the denominator.

Therefore, {(3 + √8)/(2 + √8)} × {(2 – √8) / (2 – √8)}

= {(3 + √8) (2 – √8)} / {(2 + √8 )(2 – √8)}

= (6 – 3√8 + 2√8 – 8) / {(2)2  – (√8 )2}

= (-2 – √8) / {4 – 8}

= (-2 – √8 )/(-8)

= {-(2 + √8)}/(-8)

= (2 + √8) / 8

Question 7: Rationalize the denominator and simplify, if possible. 6/(8 + √5) ?

Solution: 

Given: 6/(8 + √5)

Multiply the denominator and numerator by a conjugate that removes the radicals in the denominator.

Therefore, {6/(8 + √5)} × {(8 – √5) / (8 – √5)}

= {6 (8 – √5)} / {( 8 + √5)(8 – √5)}

= (48 – 6√5) / {(8)2  – (√5)2}

= (48 – 6√5) / {64 – 5}

= (48 – 6√5)/59

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