Evaluate: `(4a-5b)^(3)`

Answer

To evaluate \((4a - 5b)^3\), we can use the formula for the cube of a binomial, which is given by:

\[ (x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3 \]

In our case, we will let \(x = 4a\) and \(y = 5b\). Thus, we can rewrite the expression as:

\[ (4a - 5b)^3 = (4a)^3 - 3(4a)^2(5b) + 3(4a)(5b)^2 - (5b)^3 \]

Now, we will calculate each term step by step.

### Step 1: Calculate \((4a)^3\)
\[ (4a)^3 = 4^3 \cdot a^3 = 64a^3 \]

### Step 2: Calculate \(3(4a)^2(5b)\)
\[ 3(4a)^2(5b) = 3 \cdot 16a^2 \cdot 5b = 240a^2b \]

### Step 3: Calculate \(3(4a)(5b)^2\)
\[ 3(4a)(5b)^2 = 3 \cdot 4a \cdot 25b^2 = 300ab^2 \]

### Step 4: Calculate \((5b)^3\)
\[ (5b)^3 = 5^3 \cdot b^3 = 125b^3 \]

### Step 5: Combine all the terms
Now we can combine all the calculated terms:
\[ (4a - 5b)^3 = 64a^3 - 240a^2b + 300ab^2 - 125b^3 \]

Thus, the final result is:
\[ (4a - 5b)^3 = 64a^3 - 240a^2b + 300ab^2 - 125b^3 \]