Question

Question:-
Rationalise the denominator of:-
$\frac{1}{ \sqrt{7 } - 2 }$
Note:-
▪️Grade - 9
▪️Chapter - Polynomials
✅Quality answer needed
✅Step - by - step explanation required ( must )
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Answer 1

$\Huge \fbox{{\color{magenta}{\textsf{\textbf{Solution :-}}}}}$

Given :-

$\: \: \: \: \: \: \longmapsto \sf \frac{1}{ \sqrt{7} - 2 }$

Rationalising the denominator :-

Multiply the fraction by the conjugate of the denominator i.e., √7 - 2

$\: \: \: \: \: \:\longmapsto \sf \frac{1}{ \sqrt{7} - 2 } \times \frac{ \sqrt{7} + 2 }{ \sqrt{7} + 2}$

$\: \: \: \: \: \: \longmapsto \sf \frac{ \sqrt{7} + 2}{ {( \sqrt{7})}^{2} - {(2)}^{2} }$

$\: \: \: \: \: \: \longmapsto \sf \frac{ \sqrt{7} +2}{7 - 4}$

$\: \: \: \: \: \: \longmapsto \sf \frac{ \sqrt{7} +2}{3}$

★ Identity Used :-

$\large\longmapsto \sf (a-b)(a+b)=a^2-b^2$

Answer 2

Solution!!

1/(√7 - 2)

Rationalise the denominator by multiplying the fraction.

= 1/(√7 - 2) × (√7 + 2)/(√7 + 2)

= [1(√7 + 2)]/[(√7 - 2)(√7 + 2)]

Simplify the expressions using (a - b)(a + b) = a² - b².

= [1(√7 + 2)]/[(√7)² - (2)²]

= [1(√7 + 2)]/[7 - 4]

= [1(√7 + 2)]/3

Distribute 1 through the parentheses.

= (√7 + 2)/3

More identities:-

→ (a + b)² = a² + b² + 2ab

→ (a - b)² = a² + b² - 2ab

→ (a + b)(a - b) = a² - b²

→ (a + b)³ = a³ + b³ + 3ab(a + b)

→ (a - b)³ = a³ - b³ - 3ab(a - b)

→ a³ - b³ = (a - b)(a² + b² + ab)

→ a³ + b³ = (a + b)(a² + b² - ab)