Question

If `5^(2x - 1) = 5^(2x - 2) + 100`, find the value of (2x)^3x.

Answer 1

Given: `5^(2x - 1) = 5^(2x - 2) + 100`

Step-wise calculation:

1. Rewrite the equation to isolate the powers of 5 on one side:

`5^(2x - 1) - 5^(2x - 2) = 100`

2. Factor out the common term `5^(2x - 2)`:

`5^(2x - 2) (5^1 - 1) = 100`

Since 5¹ = 5, this becomes:

`5^(2x - 2) xx (5 - 1) = 100`

`5^(2x - 2) xx 4 = 100`

3. Divide both sides by 4:

`5^(2x - 2) = 100/4`

`5^(2x - 2) = 25`

4. Recognize that 25 = 5², so:

`5^(2x - 2) = 5^2`

Hence:

2x – 2 = 2

⇒ 2x = 4

⇒ x = 2

5. Now, find the value of (2x)^3x:

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

`(2x)^(3x) = 4^6`

6. Calculate (4⁶):

4⁶ = (2²)⁶ 

4⁶ = 2¹²

4⁶ = 4096