Simplify. Assume that all variables represent positive...

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Key Concepts

Perfect Square Trinomial

This concept involves recognizing a quadratic expression that can be written as the square of a binomial. A perfect square trinomial takes the form a²x² ± 2abxy + b²y² and factors into (ax ± by)². Understanding this allows us to simplify expressions and identify when an expression under a radical is actually a square, which simplifies the operation of taking a square root.

Radical Simplification

When simplifying square roots, if the radicand is a perfect square, such as (expression)², the square root simplifies to the absolute value of that expression. This use of the property ?(x²) = |x| is essential to ensure that the resulting expression is nonnegative, in accordance with the definition of square roots in the realm of real numbers.

Absolute Value

The absolute value function measures the nonnegative magnitude of a real number. When simplifying the square root of a squared expression, recognizing that the result is the absolute value of the original expression ensures the answer is valid under all conditions. In contexts where additional information about the sign of the variables is provided, the absolute value may sometimes be resolved to a simpler form.

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