To solve the expression [tex]\( \sqrt{\left(a^2 - b^2\right)^2} \)[/tex], let's go through the steps systematically.
1. Consider the Expression Inside the Square Root:
The expression inside the square root is [tex]\( \left(a^2 - b^2\right)^2 \)[/tex].
2. Simplify [tex]\( a^2 - b^2 \)[/tex]:
Recognize that [tex]\( a^2 - b^2 \)[/tex] is a standard algebraic expression known as the difference of squares. However, in this particular problem, this recognition isn't necessary for simplification; we proceed by focusing on [tex]\( (a^2 - b^2) \)[/tex] directly.
3. Square the Expression:
When [tex]\( a^2 - b^2 \)[/tex] is squared, we get:
[tex]\[
(a^2 - b^2)^2
\][/tex]
4. Take the Square Root of the Squared Expression:
The next step is to take the square root of [tex]\( (a^2 - b^2)^2 \)[/tex]:
[tex]\[
\sqrt{(a^2 - b^2)^2}
\][/tex]
5. Use the Property of Square Roots:
According to the property of square roots, [tex]\( \sqrt{x^2} = |x| \)[/tex]. Therefore:
[tex]\[
\sqrt{(a^2 - b^2)^2} = |a^2 - b^2|
\][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[
\sqrt{\left(a^2 - b^2\right)^2} = |a^2 - b^2|
\][/tex]
