Question

find the value of
$\frac{3}{ \sqrt{5} + \sqrt{2} } + \frac{7}{ \sqrt{5} - \sqrt{2} }$
$if \sqrt{5 = 2.236} and \sqrt{2 = 1.414}$
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Answer 1

Tᴏ Fɪɴᴅ :-

→ [3 / (√5 + √2) ] + [ 7 / (√5 - √2) ]

Sᴏʟᴜᴛɪᴏɴ :-

→ [3 / (√5 + √2) ] + [ 7 / (√5 - √2) ]

Taking LCM we get,

→ [ 3(√5 - √2) + 7(√5 + √2) ] / [ (√5 + √2) (√5 - √2) ]

Using (a + b)(a - b) = a² - b² in Denominator now, we get,

→ [3√5 - 3√2 + 7√5 + 7√2 ] / [ (√5)² - (√2)² ]

→ [ 10√5 + 4√2 ] / ( 5 - 2 )

→ ( 10√5 + 4√2 ) / 3

Putting Given values of √5 = 2.236 & √2 = 1.414 , we get,

→ [ 10*2.236 + 4*1.414 ] / 3

→ ( 22.36 + 5.656 ) / 3

→ (28.016) / 3

→ 9.33867 (Ans.)

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Lets Try it with Rationalizing the Both Terms Separately, & Than adding Them :-

→ [3 / (√5 + √2) ]

Rationalize

→ [3 / (√5 + √2) ] * [(√5 - √2) / (√5 - √2) ]

using (a + b)(a - b) = a² - b² in Denominator now, we get

→ [ 3(√5 - √2) ] / [ (√5)² - (√2)² ]

→ [ 3(√5 - √2) / ( 5 - 2) ]

→ [ 3(√5 - √2) / 3 ]

→ ( √5 - √2 )

Similarly, Rationalize the second part,

→ [ 7 / (√5 - √2) ]

Rationalize

→ [7 / (√5 - √2) ] * [(√5 + √2) / (√5 + √2) ]

using (a + b)(a - b) = a² - b² in Denominator now, we get

→ [ 7(√5 + √2) ] / [ (√5)² - (√2)² ]

→ [ 7(√5 + √2) / ( 5 - 2) ]

→ [ 7(√5 + √2) / 3 ]

Adding Both Parts now, we get,

→ ( √5 - √2 ) + [ 7(√5 + √2) / 3 ]

Taking LCM ,

→ [ 3(√5 - √2) + 7(√5 + √2 ] / 3

→ [ 3√5 - 3√2 + 7√5 + 7√2 ] / 3

→ [ 10√5 + 4√2 ] / 3

→ [ 10*2.236 + 4*1.414 ] / 3

→ ( 22.36 + 5.656 ) / 3

→ (28.016) / 3

→ 9.33867 (Ans.)

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Answer 2

Solution

Given:-

• √5 = 2.236
• √2 = 1.414

Find:-

• 3/(√5 + √2) + 7/(√5 - √2)

Explanation

==> 3/(√5 + √2) + 7/(√5 - √2)

Rationalize Denominators of all terms

==> [ 3(√5 - √2)/(√5 + √2)(√5 - √2) ] + [ 7(√5 + √2)/(√5 - √2)(√5 + √2) ]

Using Formula

★ (x-y)(x+y) = (x² - y²)

Then,

==> [3(√5 - √2)/{(√5)² - (√2)²} ] + [ 7(√5 + √2)/{(√5)² - (√2)²} ]

==> 3(√5 - √2)/(5 - 2) + 7(√5 + √2)/(5 - 2)

==> 3(√5 - √2)/3 + 7(√5 + √2)/3

==> (√5 - √2) + 7/3 * (√5 + √2)

Keep value of √5 = 2.236 & √2 = 1.414

==> (2.236 - 1.414) + 7/3 * (2.236 + 1.414)

==> 0.822 + 7/3 * 3.650

==> 0.822 + 25.55/3

==> 0.822 + 8.517

==> 9.339 (Only take three number after point) [ Ans]

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