Simplify 5/(4 – √3)
Last Updated :
21 Dec, 2023
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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