Question

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Answer 1

Question

Prove that:-

1/(3-√8) - 1/(√8 -√7) + 1/(√7-√6) - 1/(√6-√5) + 1/(√5 -2) = 5

Solution

Take L.H.S.

➤ 1/(3-√8) - 1/(√8 -√7) + 1/(√7-√6) - 1/(√6-√5) + 1/(√5 -2)

Rationalize denominators of all term

➤ (3+√8)/(3-√8)(3+√8) - (√8+√7)/(√8+√7)(√8 -√7) + (√7-√6)/(√7-√6)(√7-√6) - (√6+√5)/(√6+√5)(√6-√5) + (√5+2)/(√5+2)(√5 -2)

Using Formula

★ (x²-y²) = (x-y)(x+y)

➤ (3+√8)/{(3)²-(√8)²} - (√8+√7)/{(√8)²-(√7)²} + (√7-√6)/{(√7)²-(√6)²} - (√6+√5)/{(√6)²-(√5)²} + (√5+2)/{(√5)²-(2)²}

➤ (3+√8)/(9-8) - (√8+√7)/(8-7) + (√7-√6)/(7-6) - (√6+√5(/(6-5) + (√5+2)/(5-4)

➤(3+√8) - (√8+√7) + (√7-√6) - (√6+√5) + (√5 + 2)

➤ (3+2)+(√8-√8)+(√7-√7)+(√6-√6)+√5-√5)

➤ 5 + 0 + 0 + 0 + 0

➤ 5

= R.H.S

That's proved.

____________________

Answer 2

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Prove that:-

1/(3-√8) - 1/(√8 -√7) + 1/(√7-√6) - 1/(√6-√5) + 1/(√5 -2) = 5

Solution

Take L.H.S.

➤ 1/(3-√8) - 1/(√8 -√7) + 1/(√7-√6) - 1/(√6-√5) + 1/(√5 -2)

Rationalize denominators of all term

➤ (3+√8)/(3-√8)(3+√8) - (√8+√7)/(√8+√7)(√8 -√7) + (√7-√6)/(√7-√6)(√7-√6) - (√6+√5)/(√6+√5)(√6-√5) + (√5+2)/(√5+2)(√5 -2)

Using Formula

★ (x²-y²) = (x-y)(x+y)

➤ (3+√8)/{(3)²-(√8)²} - (√8+√7)/{(√8)²-(√7)²} + (√7-√6)/{(√7)²-(√6)²} - (√6+√5)/{(√6)²-(√5)²} + (√5+2)/{(√5)²-(2)²}

➤ (3+√8)/(9-8) - (√8+√7)/(8-7) + (√7-√6)/(7-6) - (√6+√5(/(6-5) + (√5+2)/(5-4)

➤(3+√8) - (√8+√7) + (√7-√6) - (√6+√5) + (√5 + 2)

➤ (3+2)+(√8-√8)+(√7-√7)+(√6-√6)+√5-√5)

➤ 5 + 0 + 0 + 0 + 0

➤ 5

= R.H.S

That's proved

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