Question

simplify \\(\sqrt{a^2}\\) ?

Explanation:

Step1: Recall the property of square roots

The square root of a square of a real number \( a \) is related to the absolute value. The formula for \( \sqrt{x^2} \) is \( |x| \) because the square root function gives the non - negative root, and squaring a number (whether positive or negative) and then taking the square root should result in a non - negative number that has the same magnitude as the original number. For \( \sqrt{a^2} \), we apply this property.

Step2: Apply the property to \( \sqrt{a^2} \)

Using the property \( \sqrt{x^2}=|x| \), when \( x = a \), we get \( \sqrt{a^2}=|a| \). The absolute value \( |a| \) is defined as \( a \) when \( a\geq0 \) and \( - a \) when \( a < 0 \), which can be written as \(|a|=

$$\begin{cases}a, & a\geq0\\-a, & a < 0\end{cases}$$

\)

Answer:

\(\vert a\vert\) (or equivalently \(

$$\begin{cases}a, & a\geq0\\-a, & a < 0\end{cases}$$

\))