Question

if a=5+2√6 and b=1/a then what will be the value of a²+b² and a³+b³​

Answer 1

Given that :

a = 5 + 2√6.. (1)

b = 1/a.. (2)

Putting the value of a in eq. (2)

Rationalizing the denominator,

=1/( 5 + 2√6)

=> (5 - 2√6)/(5 + 2√6)(5 - 2√6)

=> (5 - 2√6)/[(5)² - (2√6)²]

=> (5 - 2√6)/(25 - 24)

=> 5 - 2√6

Now,

To find : a²+b²

We know that,

a² + b² = (a + b)² - 2ab

=> (5 + 2√6 + 5 - 2√6)² - 2(5 + 2√6)(5 - 2√6)

=(10)² - 2 × (25 -24)

=100 - 2 × 1

= 98

Thus, Value of a² + b² = 98.

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Answer 2

Answer:

$a^{2} + b^{2} = 98\\\\a^{3} + b^{3} = 970$

Step-by-step explanation:

Given:

$a = 5 + 2\sqrt{6}$

$b = \frac{1}{a}\\\\= > b = \frac{1}{5 + 2\sqrt{6} }$

Rationalizing the denominator,

$= > b = \frac{1}{5 + 2\sqrt{6} } \times \frac{5 - 2\sqrt{6} }{5 - 2\sqrt{6} } \\\\= > b = \frac{5 - 2\sqrt{6} }{(5 +2 \sqrt{6} )(5 - 2\sqrt{6} )} \\\\= > b = \frac{5 - 2\sqrt{6} }{(5)^{2} - (2\sqrt{6}) ^{2} } [since, (a+b)(a-b)= a^{2} - b^{2} ] \\\\= > b = \frac{5 - 2\sqrt{6} }{25 - 24} [since, (2\sqrt{6})^{2} = 2^{2} \times (\sqrt{6})^{2} = 4 \times 6 = 24 ]\\\\= > b = \frac{ 5 - 2 \sqrt{6} }{1} \\= > b = 5 - 2 \sqrt{6}$

Now,

$a^{2} + b^{2} = (5 + 2\sqrt{6}) ^{2} + (5 - 2\sqrt{6}) ^{2}\\\\= (5^{2} + (2\times5\times2\sqrt{6}) + (2\sqrt{6} )^{2} ) + (5^{2} -(2\times5\times2\sqrt{6}) + (2\sqrt{6} )^{2})\\\\= 5^{2} + 5^{2} + (2\sqrt{6} )^{2} + (2\sqrt{6} )^{2} \\\\= 25 + 25 + 24 + 24\\\\= 98$

Algebraic identities used here:

$(a+b)^{2} = a^{2} + 2ab + b^{2} \\(a-b)^{2} = a^{2} - 2ab + b^{2}$

Next,

$a^{3}+b^{3} = (5+2\sqrt{6} )^{3} + (5-2\sqrt{6} )^{3}\\\\= [5^{3} + (3\times 5^{2}\times 2\sqrt{6}) + (3\times 5\times (2\sqrt{6})^{2}) + (2\sqrt{6}) ^{3}] \\\\+ [5^{3} - (3\times 5^{2}\times 2\sqrt{6}) + (3\times 5\times (2\sqrt{6})^{2}) - (2\sqrt{6}) ^{3}]\\\\= 5^{3} + 5^{3} + (3\times 5\times (2\sqrt{6})^{2}) + (3\times 5\times (2\sqrt{6})^{2})\\\\= 125 + 125 + 360 + 360\\\\= 970$

Algebraic identities used here:

$(a+b)^{3} = a^{3} + 3a^{2} b + 3ab^{2} + b^{3} \\(a-b)^{3} = a^{3} - 3a^{2} b + 3ab^{2} - b^{3}$

Learn more about algebraic identities:

https://brainly.in/question/9623615

Similar problems:

https://brainly.in/question/6521317