Question

If a =8+3√7 & b = 1/a. What will be the value of a square +b square?? Pls help me pls it's urgent ​

Answer 1

Answer:

The required answer is 254.

Step-by-step explanation:

Given :

a = 8 + 3√7

b = 1/a

To find :

the value of (a² + b²)

Solution :

We know,

(a + b)² = a² + b² + 2ab

a² + b² = (a + b)² - 2ab

a² + b² = (a + b)² - 2(a)(1/a)

a² + b² = (a + b)² - 2

So, we have to find out the value of (a + b)².

$\tt b = \dfrac{1}{a} \\ \tt b = \dfrac{1}{8+3\sqrt{7}}$

We need to rationalize the denominator.

Rationalizing factor = 8 – 3√7

Multiply and divide the fraction by the rationalizing factor.

$\tt b = \dfrac{1}{8+3\sqrt{7}} \times \dfrac{8-3\sqrt{7}}{8-3\sqrt{7}} \\\\ \tt b = \dfrac{8-3\sqrt{7}}{(8+3\sqrt{7})(8-3\sqrt{7})} \\\\ \tt b = \dfrac{8-3\sqrt{7}}{8^2-(3\sqrt{7})^2} [ \because (x+y) (x-y) = x² - y²) ] \\\\ \tt b = \dfrac{8-3\sqrt{7}}{64 - 9(7)} \\\\ \tt b = \dfrac{8-3\sqrt{7}}{64-63} \\\\ \tt b = \dfrac{8-3\sqrt{7}}{1} \\\\ \longrightarrow \tt b = 8-3√7$

Now,

(a + b)² = (8 + 3√7 + 8 - 3√7)²

(a + b)² = (16)²

(a + b)² = 256

________________________

(a² + b²) = (a + b)² - 2

= 256 - 2

= 254

The required answer is 254.