Let's solve the problems step by step.

**BT1: So sánh (Comparison)**

a) \(4^{12}\) và \(8^8\)

First, express both numbers with the same base:
\[4 = 2^2 \Rightarrow 4^{12} = (2^2)^{12} = 2^{24}\]
\[8 = 2^3 \Rightarrow 8^8 = (2^3)^{8} = 2^{24}\]

So, \(4^{12} = 8^8\).

b) \(3^{15}\) và \(27^5\)

Express both numbers with the same base:
\[27 = 3^3 \Rightarrow 27^5 = (3^3)^5 = 3^{15}\]

So, \(3^{15} = 27^5\).

c) \(5^{30}\) và \(10^{20}\)

Express both numbers with the same base:
\[10 = 2 \times 5\]
\[10^{20} = (2 \times 5)^{20} = 2^{20} \times 5^{20}\]

Since \(5^{30}\) and \(5^{20}\) have different exponents, we need to compare:
\[5^{30} \text{ and } 2^{20} \times 5^{20}\]

Clearly, \(5^{30} > 2^{20} \times 5^{20}\).

d) \((-32)^9\) và \((-16)^{12}\)

Express both numbers with the same base:
\[-32 = -2^5 \Rightarrow (-32)^9 = (-2^5)^9 = -2^{45}\]
\[-16 = -2^4 \Rightarrow (-16)^{12} = (-2^4)^{12} = -2^{48}\]

Since \(-2^{45}\) and \(-2^{48}\) have different exponents, we need to compare:
Clearly, \(-2^{45} > -2^{48}\).

**BT2:**

\[S = 1 + 2 + 2^2 + 2^3 + \ldots + 2^{2023}\]

This is a geometric series with the first term \(a = 1\) and common ratio \(r = 2\). The sum of the first \(n\) terms of a geometric series is given by:
\[S_n = a \frac{r^n - 1}{r - 1}\]

Here, \(n = 2024\):
\[S = \frac{2^{2024} - 1}{2 - 1} = 2^{2024} - 1\]

**BT3: Tìm x biết (Find x)**

a) \((2x - x)^2 = 25\)

Simplify:
\[x^2 = 25\]
\[x = \pm 5\]

b) \((3x - 4)^3 = 125\)

Take the cube root of both sides:
\[3x - 4 = 5\]
\[3x = 9\]
\[x = 3\]

c) \((2x - 1)^6 = (2x - 1)^8\)

Since the bases are the same, the exponents must be equal:
\[6 = 8\]
This is a contradiction, so there is no solution.

d) \(5^x + 5^{x+2} = 650\)

Express \(5^{x+2}\) in terms of \(5^x\):
\[5^x + 25 \cdot 5^x = 650\]
\[26 \cdot 5^x = 650\]
\[5^x = 25\]
\[x = 2\]

**BT4: Chứng minh rằng (Prove that)**

a) \(10^6 - 57 = 59\)

This is incorrect. The correct statement should be verified.

b) \(87 - 22 \cdot 18 = 14\)

Calculate:
\[87 - 396 = -309\]

This is incorrect. The correct statement should be verified.

c) \(10^7 + 10^8 + 10^9 = 222\)

Calculate:
\[10^7 + 10^8 + 10^9 = 10000000 + 100000000 + 1000000000 = 1110000000\]

This is incorrect. The correct statement should be verified.

The given statements in BT4 are incorrect and need to be rechecked.