What is the HCF of `a^2b^4+2a^2b^2 `and `(ab)^7 - 4a^2b^9`

Options

A. `ab`
B. `a^2b^3`
C. ` a^2b^2`
D. `a^3b^2`

Answer

To find the HCF (Highest Common Factor) of the expressions \( a^2b^4 + 2a^2b^2 \) and \( (ab)^7 - 4a^2b^9 \), we will follow these steps:

### Step 1: Factor the first expression
The first expression is \( a^2b^4 + 2a^2b^2 \).
- We can see that \( a^2b^2 \) is common in both terms.
- Factoring out \( a^2b^2 \), we get:
\[ a^2b^2(b^2 + 2) \]

### Step 2: Factor the second expression
The second expression is \( (ab)^7 - 4a^2b^9 \).
- We can rewrite \( (ab)^7 \) as \( a^7b^7 \).
- The second term \( 4a^2b^9 \) can be factored as \( 4a^2b^9 \).
- The common factor in both terms is \( a^2b^7 \).
- Factoring out \( a^2b^7 \), we get:
\[ a^2b^7(a^5 - 4b^2) \]

### Step 3: Identify the common factors
Now we have:
1. From the first expression: \( a^2b^2(b^2 + 2) \)
2. From the second expression: \( a^2b^7(a^5 - 4b^2) \)

The common factors between these two expressions are \( a^2 \) and \( b^2 \).

### Step 4: Determine the HCF
To find the HCF, we take the lowest powers of the common factors:
- For \( a \), the minimum power is \( a^2 \).
- For \( b \), the minimum power is \( b^2 \).

Thus, the HCF is:
\[ \text{HCF} = a^2b^2 \]

### Final Answer
The HCF of \( a^2b^4 + 2a^2b^2 \) and \( (ab)^7 - 4a^2b^9 \) is \( a^2b^2 \).

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