Question

(4+√15)^x + (4-√15)^x =8​

Answer 1

Answer:

i) 1/(4 + √15) = (4 - √15)/{(4 + √15)*(4 - √15)} = (4 - √15)/(16 - 15) = 4 - √15

ii) Thus the given expression is: (4 + √15)^x + {1/(4 + √15)}^x = 8

iii) Let y = (4 + √15)^x

So the given expression is: y + 1/y = 8

Rearranging, y² - 8y + 1 = 0

Solving the above using quadratics formula, y = 4 ± √15

Thus y = (4 + √15)^x = (4 + √15)

So x = 1 (equating the powers, as base is same on both sides)

Similarly considering the other value,

y = (4 + √15)^x = (4 - √15) = 1/(4 + √15) = (4 + √15)^(-1)

So again equating the powers, x = -1

Thus x = ±1

Checking the same with the given expression, both values satisfies the given equation.

So our value of x = ±1 is correct.

Answer 2

Answer:

please view given images for the answer