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Chapter 5: Problem 51

Find each product. $$\left(x^{2}-2 y^{2}\right)^{2}$$

Short Answer

Expert verified
The product of \((x^{2}-2 y^{2})^{2}\) is \(x^{4} - 4x^{2}y^{2} + 4y^{4}\).

Step by step solution

01

Expanding the Term

The task is to expand \((x^{2}-2 y^{2})^{2}\) which is in the form of \((a-b)^2\). To do this, we will use the general formula \((a-b)^2 = a^2 - 2ab + b^2\). Here, \(a = x^2\) and \(b= 2y^2\). Let's substitute these into the formula.
02

Substitution Part 1

Substitute \(a\) with \(x^2\) and \(b\) with \(2y^2\) so we have \((a - b)^2\) become \((x^{2}-2 y^{2})^{2}\) equals to \((x^{2})^{2} - 2*(x^{2})*(2y^{2}) + (2y^{2})^{2}\).
03

Substitution Part 2

Continue by evaluating each term individually. \((x^{2})^{2} = x^{4}\), \(2*(x^{2})*(2y^{2}) = 4x^{2}y^{2}\) and \((2y^{2})^{2} = 4y^{4}\).
04

Final Step

Finally, substitute the results back into the expression for the expanded form: \(x^{4} - 4x^{2}y^{2} + 4y^{4}\).

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Most popular questions from this chapter

Are the expressions $$ \frac{12 x^{2}+6 x}{3 x} \text { and } 4 x+2 $$ equal for every value of \(x ?\) Explain.

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