Question

solve q.15
correct answer only​

Answer 1

Step-by-step explanation:

here is your answer and once try to solve it yourself

Answer 2

Question:

If a = ${\sf{3 + 2 {\sqrt{2}} , }}$ find the value of ${\sf{a^2 - {\dfrac{1}{a^2}} .}}$

Answer:

- 24√2

Step-by-step explanation:

Given : ${\sf{\ \ a = 3 + 2 {\sqrt{2}} }}$

To Find : ${\sf{\ \ a^2 - {\dfrac{1}{a^2}}}}$

Solution :

First, we find out ${\sf{ {\dfrac{1}{a}} .}}$

$\implies{\sf{ {\dfrac{1}{a}} = {\dfrac{1}{3 - 2 {\sqrt{2}} }} }}$

• Rationalising the denominator.

$\implies{\sf{ {\dfrac{1}{a}} = {\dfrac{1}{3 - 2 {\sqrt{2}} }} \times {\dfrac{3 + 2 {\sqrt{2}} }{ 3 + 2 {\sqrt{2}}}}}}$

$\implies{\sf{ {\dfrac{1}{a}} = {\dfrac{3 + 2 {\sqrt{2}} }{ (3 - 2 {\sqrt{2}} )( 3 + 2 {\sqrt{2}} )}}}}$

${\boxed{\tt{\bullet \ Identity \ : \ (a - b)(a + b) = a^2 - b^2}}}$

${\tt{\quad Here, \ a = 3, \ b = 2 {\sqrt{2}} }}$

$\implies{\sf{ {\dfrac{1}{a}} = {\dfrac{3 + 2 {\sqrt{2}} }{ (3)^2 - (2 {\sqrt{2}} )^2}}}}$

$\implies{\sf{ {\dfrac{1}{a}} = {\dfrac{3 + 2 {\sqrt{2}} }{9 - 8}}}}$

$\implies{\sf{ {\dfrac{1}{a}} = 3 + 2 {\sqrt{2}} }}$

_______________________________

Second, we find out a².

$\implies{\sf{a^2 = (3 - 2 {\sqrt{2}} )^2 }}$

${\boxed{\tt{\bullet \ Identity \ : \ (a - b)^2 = a^2 + b^2 - 2ab}}}$

${\tt{\quad Here, \ a = 3, \ b = 2 {\sqrt{2}} }}$

$\implies{\sf{a^2 = (3)^2 + (2 {\sqrt{2}} )^2 - 2(3)(2 {\sqrt{2}} )}}$

$\implies{\sf{a^2 = 9 + 8 - 12 {\sqrt{2}}}}$

$\implies{\sf{a^2 = 17 - 12 {\sqrt{2}}}}$

_______________________________

Third, we find out ${\sf{ {\dfrac{1}{a^2}} .}}$

$\implies{\sf{ {\dfrac{1}{a^2}} = (3 + 2 {\sqrt{2}} )^2}}$

${\boxed{\tt{\bullet \ Identity \ : \ (a + b)^2 = a^2 + b^2 + 2ab}}}$

${\tt{\quad Here, \ a = 3, \ b = 2 {\sqrt{2}} }}$

$\implies{\sf{ {\dfrac{1}{a^2}} = (3)^2 + (2 {\sqrt{2}} )^2 + 2(3)(2 {\sqrt{2}} )}}$

$\implies{\sf{ {\dfrac{1}{a^2}} = 9 + 8 + 12 {\sqrt{2}} }}$

$\implies{\sf{ {\dfrac{1}{a^2}} = 17 + 12 {\sqrt{2}} }}$

_______________________________

Fourth, we find out ${\sf{a^2 - {\dfrac{1}{a^2}} .}}$

$\implies{\sf{a^2 - {\dfrac{1}{a^2}} = (17 - 12 {\sqrt{2}} ) - (17 + 12 {\sqrt{2}} )}}$

$\implies{\sf{a^2 - {\dfrac{1}{a^2}} = 17 - 12 {\sqrt{2}} - 17 - 12 {\sqrt{2}}}}$

$\implies{\sf{a^2 - {\dfrac{1}{a^2}} = - 12 {\sqrt{2}} - 12 {\sqrt{2}} }}$

$\implies{\boxed{\boxed{\sf{a^2 - {\dfrac{1}{a^2}} = - 24 {\sqrt{2}} }}}}$