Question

3 power X+ 3 power -x=82/9​

Answer 1

Ans ---> x = -2 or 2

Given ---> 3ˣ + 3⁻ˣ = 82 / 9

To find---> Value of x

Solution ---> ATQ,

3ˣ + 3⁻ˣ = 82 / 9

Let , 3ˣ = y

=> y + y⁻¹ = 82 / 9

Multiplying whole equation by y , we get

=> y × y + y × y⁻¹ = ( 82 / 9 ) y

We have a law of exponent,

aᵐ aⁿ = aᵐ⁺ⁿ , applying it here we get

=> y¹⁺¹ + y¹⁻¹ = 82 y / 9

=> y² + y⁰ = 82 y / 9

We have a law of exponent

a⁰ = 1 , applying this here , we get

=> y² + 1 = 82 y / 9

Multiplying whole equation by 9 , we get

=> 9 y² + 9 = 82 y

=> 9 y² - 82y + 9 = 0

It is a quadratic equation , applying method of splitting the middle term , we get

=> 9 y² - ( 81 + 1 ) y + 9 = 0

=> 9 y² - ( 81y + y ) + 9 = 0

=> 9 y² - 81y - y + 9 = 0

=> 9y ( y - 9 ) - 1 ( y - 9 ) = 0

=> ( y - 9 ) ( 9 y - 1 ) = 0

If y - 9 = 0

=> y = 9

Putting y = 3ˣ in it we get

=> 3ˣ = 9

=> 3ˣ = 3²

Comparing exponents from both sides , we get

=> x = 2

If 9 y - 1 = 0

=> 9y = 1

=> y = 1 / 9

=> y = 1 / 3²

Putting y = 3ˣ in it we get

=> 3ˣ = 3⁻²

On comparing exponents we get

=> x = -2

So x = -2 or 2