The expression in the image is $(a - 2b)^3$. Let's expand it using the binomial theorem:

Binomial Expansion Formula:

$(x + y)^n = \sum_{k=0}^n \binom{n}{k} x^{n-k} y^k$

Here, $a - 2b$ is the binomial, and $n = 3$. Substituting $x = a$ and $y = -2b$, we expand:

$(a - 2b)^3 = \binom{3}{0} a^3 (-2b)^0 + \binom{3}{1} a^2 (-2b)^1 + \binom{3}{2} a^1 (-2b)^2 + \binom{3}{3} a^0 (-2b)^3$

Step-by-step Calculation:

1. $\binom{3}{0} a^3 (-2b)^0 = 1 \cdot a^3 \cdot 1 = a^3$
2. $\binom{3}{1} a^2 (-2b)^1 = 3 \cdot a^2 \cdot (-2b) = -6a^2b$
3. $\binom{3}{2} a^1 (-2b)^2 = 3 \cdot a \cdot (4b^2) = 12ab^2$
4. $\binom{3}{3} a^0 (-2b)^3 = 1 \cdot 1 \cdot (-8b^3) = -8b^3$

Final Expanded Form:

$(a - 2b)^3 = a^3 - 6a^2b + 12ab^2 - 8b^3$

Would you like a detailed explanation of any step or further examples? 😊

Related Questions:

1. What is the binomial expansion for $(a + 2b)^3$?
2. How does the expansion change if the exponent is 4?
3. What happens when $b = 0$ in the expansion of $(a - 2b)^3$?
4. How is Pascal's triangle used in binomial expansion?
5. Can you derive the general term in the expansion of $(a - 2b)^n$?

Tip:

When expanding binomials, remember to carefully apply the signs for negative terms, as errors often occur there!