91Ó°ÊÓ

In mathematics, the absolute value of a number is its distance from zero on the number line, regardless of direction. Because distance is always non-negative, the absolute value is always zero or positive.
The absolute value is denoted by two vertical bars, such as \(|a|\). For example:

• \( |3| = 3 \)
• \( |-3| = 3 \)

It is crucial to apply absolute value when dealing with even roots of even exponents. For instance, when you simplify \( \sqrt{(x^5)^2} \), you must include absolute value, resulting in \(|x^5|\). This addresses the potential for both positive and negative values of \(x\).

Radicals, or roots, have specific properties that help in simplification. One important property is:

• \( \sqrt{a^2} = |a| \): The square root of a squared number results in the absolute value of the base.

Other properties include:

• \( \sqrt{a \cdot b} = \sqrt{a} \cdot \sqrt{b} \): The square root of a product is the product of the square roots.
• \( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \): The square root of a quotient is the quotient of the square roots.

By using these properties, simplifying radicals becomes more straightforward. In the exercise, we used \( \sqrt{x^{10}} = \sqrt{(x^5)^2} \), which simplifies to \( |x^5| \).

Even exponents have specific behaviors that are consistent and help with simplifying expressions. When a base is raised to an even exponent, the result is always non-negative. This is because multiplying a number by itself an even number of times always results in a positive product, even if the base was negative.
For example:

• \( (-2)^2 = 4 \)
• \( (-2)^4 = 16 \)
• \( (2^2) = 4 \)
• \( (2^4) = 16 \)

In the context of the exercise, the expression \( x^{10} \) means we are multiplying \( x \) by itself ten times. When simplifying \( \sqrt{x^{10}} \), recognizing it as \( \sqrt{(x^5)^2} \) allows us to use the property of radicals, ensuring the simplified form is the absolute value \( |x^5| \). This approach helps to manage both positive and negative values for \( x \).