Expand : (ii) `(5y - 3x)^3`

Answer

To expand the expression \((5y - 3x)^3\), we can use the identity for the cube of a binomial, which is given by:

\[ (a - b)^3 = a^3 - b^3 - 3ab(a - b) \]

In our case, we can identify:
- \(a = 5y\)
- \(b = 3x\)

Now, we will substitute these values into the identity.

### Step 1: Substitute \(a\) and \(b\) into the identity

Using the identity, we have:

\[ (5y - 3x)^3 = (5y)^3 - (3x)^3 - 3(5y)(3x)(5y - 3x) \]

### Step 2: Calculate \(a^3\) and \(b^3\)

Now, we calculate \(a^3\) and \(b^3\):

\[ (5y)^3 = 125y^3 \]
\[ (3x)^3 = 27x^3 \]

### Step 3: Calculate \(3ab\)

Next, we calculate \(3ab\):

\[ 3(5y)(3x) = 45xy \]

### Step 4: Substitute back into the expression

Now, we substitute these values back into the expression:

\[ (5y - 3x)^3 = 125y^3 - 27x^3 - 45xy(5y - 3x) \]

### Step 5: Expand the last term

Now we need to expand the last term \( - 45xy(5y - 3x) \):

\[ - 45xy(5y) + 45xy(3x) = -225xy^2 + 135x^2y \]

### Step 6: Combine all the terms

Putting everything together, we have:

\[ (5y - 3x)^3 = 125y^3 - 27x^3 - 225xy^2 + 135x^2y \]

### Final Answer

Thus, the final expanded form of \((5y - 3x)^3\) is:

\[ 125y^3 - 27x^3 - 225xy^2 + 135x^2y \]

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