Square Roots 080152

1. **Problem Statement:** Understand what a square root is and how to work with it. 2. **Definition:** The square root of a number $x$ is a value $y$ such that $$y^2 = x$$. 3. **Important Rules:** - The square root symbol is $\sqrt{\cdot}$. - For any positive number $a$, $\sqrt{a}$ is the positive number whose square is $a$. - $\sqrt{a^2} = |a|$, the absolute value of $a$. 4. **Examples:** - $\sqrt{9} = 3$ because $3^2 = 9$. - $\sqrt{16} = 4$ because $4^2 = 16$. 5. **Simplifying Square Roots:** - Factor the number inside the root into prime factors. - Pair the factors and bring one from each pair outside the root. Example: Simplify $\sqrt{50}$. $$\sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2}$$ 6. **Negative Numbers:** - The square root of a negative number is not a real number. - It involves imaginary numbers, e.g., $\sqrt{-1} = i$. 7. **Summary:** - Square roots undo squaring. - They are always positive or zero for real numbers. - Simplify by factoring inside the root. This explanation covers the basics of square roots.