rationalize the denominator of 30/ki
5√3-3√5
Answers
Answer:
Here is your answer:
To rationalize the expression:
\frac{30}{5 \sqrt{3} -3 \sqrt{5} }
5
3
−3
5
30
Solution:
To rationalize a number we must multiply and divide the number by it's conjugate.
Conjugate of denominator is 5√3 + 3√5.
Then,
\frac{30}{5 \sqrt{3} -3 \sqrt{5} }= \frac{30}{5 \sqrt{3} -3 \sqrt{5} }\times \frac{5 \sqrt{3} +3 \sqrt{5}}{5 \sqrt{3} +3 \sqrt{5}}
5
3
−3
5
30
=
5
3
−3
5
30
×
5
3
+3
5
5
3
+3
5
We know that,
( a+b)(a-b) =a^{2}-b^{2}(a+b)(a−b)=a
2
−b
2
Using the identity to multiply the denominators , We get.
(5 \sqrt{3} -3 \sqrt{5})(5 \sqrt{3} +3 \sqrt{5}) = (5 \sqrt{3})^{2} -(3 \sqrt{5})^{2} = 75 -45 = 30(5
3
−3
5
)(5
3
+3
5
)=(5
3
)
2
−(3
5
)
2
=75−45=30
Then,
\frac{30}{5 \sqrt{3} -3 \sqrt{5} }\times \frac{5 \sqrt{3} +3 \sqrt{5}}{5 \sqrt{3} +3 \sqrt{5}} = \frac{(30)(5 \sqrt{3} +3 \sqrt{5}) }{30} = \frac{5 \sqrt{3} -3 \sqrt{5} }{1} =5 \sqrt{3} -3 \sqrt{5}
5
3
−3
5
30
×
5
3
+3
5
5
3
+3
5
=
30
(30)(5
3
+3
5
)
=
1
5
3
−3
5
=5
3
−3
5
Hence the denominator is rationalized.