Math Editor

Square Roots

1. 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.

2. 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).

3. Range: The output of √x is always non-negative.

• • So, √x≥0.

4. 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∑5​πnn​)​ is equal to: ________________________
∫0∞​√π×√xxππ​​dx is equal to: ____________________________