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.