Question

Solve the following equation:

`3^(2x + 1) = 3^(2x - 1) + 216`

Answer 1

Given equation: `3^(2x + 1) = 3^(2x - 1) + 216`

Step-wise calculation:

Step 1: Rewrite the powers of 3 for simplification

Note that:

`3^(2x + 1) = 3^(2x) xx 3^1`

`3^(2x + 1) = 3 xx 3^(2x)`

And `3^(2x - 1) = 3^(2x)/3`

`3^(2x - 1) = 3^(2x) xx 3^-1`

Substitute these back into the equation:

`3 xx 3^(2x) = 3^(2x)/3 + 216`

Step 2: Let (y = 3^2x), then rewrite the equation:

`3y = y/3 + 216`

Multiply through by 3 to clear the denominator:

9y = y + 648

Step 3: Solve for (y): 

9y – y = 648 

⇒ 8y = 648

⇒ `y = 648/8`

⇒ y = 81

Recall that:

y = 3^2x

y = 81

Step 4: Express 81 as a power of 3: 

81 = 3⁴

So, 3^2x = 3⁴ 

⇒ 2x = 4

⇒ x = 2