The square root of a number is a value that, when multiplied by itself, gives the original number. It is denoted by the radical symbol \( \sqrt{} \). For example, \( \sqrt{9} = 3 \) because \( 3 \times 3 = 9 \).
Key Properties
\( \sqrt{a^2} = |a| \), for any real number \( a \)
\( \sqrt{ab} = \sqrt{a} \cdot \sqrt{b} \), for non-negative \( a \) and \( b \)
\( \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} \), where \( b \neq 0 \)
The square root of a negative number is not a real number (e.g., \( \sqrt{-1} = i \), the imaginary unit)
Applications
Used in geometry to calculate lengths using the Pythagorean theorem.
Appears in solving quadratic equations and algebraic simplifications.
Common in science and engineering for modeling areas, speeds, and signal strengths.
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