Evaluate:
$(2 a - b) (2 a + b) (4 a^{2} + b^{2})$ is $16 a^{4} - b^{4}$ .
If true, then enter $1$ and if false, then enter $0$ .

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Solution

Given, $(2 a - b) (2 a + b) (4 a^{2} + b^{2})$ .

We know, $(a + b) (a - b)$ $= (a^{2} - b^{2})$ .

Multiply by using direct method, we get,
$(2 a - b) (2 a + b) (4 a^{2} + b^{2}) = [(2 a)^{2} - b^{2}] (4 a^{2} + b^{2})$
$= (4 a^{2} - b^{2}) (4 a^{2} + b^{2})$ $= (4 a^{2} + b^{2}) (4 a^{2} - b^{2})$ .
Now, by using formula $(a^{2} - b^{2}) = (a + b) (a - b)$ , we get,
$= (((4 a)^{2})^{2} - (b^{2})^{2})$
$= 16 a^{4} - b^{4}$ .

Hence, the statement is true.
Therefore, $1$ is the correct answer.

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