Question

rationalize the denominator
$1 \div 8 + 3 \sqrt{5}$
​

Answer 1

GIVEN TO RATIONALIZE :-

$\dfrac{1}{8 + 3 \sqrt{5} }$

SOLUTION :-

Rationalizing the denominator means in denominator surds must be removed For removing this surds we have to do the Rationalizing For Rationalizing the denominator We have to multiply and divide with its Rationalizing factor

Rationalizing factor means just we have to change the sign for its denominator Since,

Rationalizing factor of

$8 + 3 \sqrt{5} \: is \: 8 - 3 \sqrt{5}$

So, multiply and divide with that

$\dfrac{1}{8 + 3 \sqrt{5} } \times \dfrac{8 - 3 \sqrt{5} }{8 - 3 \sqrt{5} }$

$\dfrac{1 \times 8 - 3 \sqrt{5} }{(8 + 3 \sqrt{5})(8 - 3 \sqrt{5} ) }$

Denominator is in form of

(a + b) (a -b) = a²- b²

$\dfrac{8 - 3 \sqrt{5} }{(8) {}^{2} - (3 \sqrt{5}) {}^{2} }$

$\dfrac{8 - 3 \sqrt{5} }{64 - 45}$

$\dfrac{8 - 3 \sqrt{5} }{19}$

Hence surds removed in denominator Since we can say that Its Rationalized

_________________

Know more Algebraic Identities:-

(a+ b)² = a² + b² + 2ab

( a - b )² = a² + b² - 2ab

( a + b )² + ( a - b)² = 2a² + 2b²

( a + b )² - ( a - b)² = 4ab

( a + b + c )² = a² + b² + c² + 2ab + 2bc + 2ca

a² + b² = ( a + b)² - 2ab

(a + b )³ = a³ + b³ + 3ab ( a + b)

( a - b)³ = a³ - b³ - 3ab ( a - b)

If a + b + c = 0 then a³ + b³ + c³ = 3abc

Answer 2

Answer:

$⟹ \tt{\frac{8 - 3 \sqrt{5} }{19} \: \: Ans. }$

Step-by-step explanation:

Given that:-

$\tt{ \frac{1}{8 + 3 \sqrt{5} } }$

What to do:-

To rationalize the denominator.

Solution:-

we have,

$\tt{ \frac{1}{8 + 3 \sqrt{5} } }$

The denominator is 8+3√5. Multiplying the numerator and denominator by 8+3√5,

We get

$\tt{ \frac{1}{8 + 3 \sqrt{5} } }$

$⟹ \tt{ \frac{1}{8 + 3 \sqrt{5} } } \times \frac{8 - 3 \sqrt{5} }{8 - 3 \sqrt{5} }$

$⟹ \frac{1}{(8 + 3 \sqrt{5}) } \times \frac{(8 - 3 \sqrt{5} )}{(8 - 3 \sqrt{5} )}$

$\tt{⬤Applying \: identity} \\ \tt \red{(a + b)(a - b) = {a}^{2} - {b}^{2} \: to \: the \: denominator}$

$⟹ \frac{8 - 3 \sqrt{5} }{( {8})^{2} - (3 \sqrt{8} {)}^{2} }$

$⟹ \frac{8 - 3 \sqrt{5} }{64 - 45}$

$⟹ \tt{\frac{8 - 3 \sqrt{5} }{19} \: \: Ans. }$

$\tt \red{Now \: the \: denominator \: is \: rationalised.}$

Hope it's help you....☺