Square Roots
- Definition: If y=√x, it means that y2=x and y≥0.
- For example, √9=3 because 32=9 and 3≥0.
- Even though (−3)2=9, √9 is not −3. The square root symbol √ specifically denotes the non-negative root.
- Domain: For √x to be a real number, the value inside the square root (the radicand) must be non-negative.
- Therefore, x≥0.
- If x<0, √x is an imaginary number (e.g., √−1=i).
- Range: The output of √x is always non-negative.
- Properties:
- Squaring: (√x)2=x for x≥0.
- Square of a number: √x2=∣x∣. This is important because x could be negative. For example, √(−3)2=√9=3=∣−3∣.
- Product Rule: √ab=√a√b for a≥0,b≥0.
- Quotient Rule: √ba=√b√a for a≥0,b>0.
- Exponent Form: √x=x1/2. This connects square roots to fractional exponents.
In summary, √x asks "what non-negative number, when multiplied by itself, gives x?"
Now, for any mathematicians reading this out there, i've left some challenges here for you. Scroll to awnser.
√85 ×√eπ is equal to: _____________
ϕ⎷π(n=1∑5πnn) is equal to: ________________________
∫0∞√π×√xxππdx is equal to: ____________________________