Square Roots Study Notes

Basic Concepts

Perfect Square: A number multiplied by itself.
- Example: 10 x 10 = 100, so 10 is the square root of 100.
Square Root: Denoted by the radical sign (\sqrt{}), represents the principal (nonnegative) root.
Radicand: The expression under the radical sign.
Negative Numbers: The square root of a negative number is not a real number.
- Example: (\sqrt{-25}) has no real number solution, but (-\sqrt{25} = -5).
Square Root and Radicand Relationship: A square root multiplied by itself equals the radicand.
- Example: (\sqrt{4} \times \sqrt{4} = 4).

(1^2 = 1)
(2^2 = 4)
(3^2 = 9)
(4^2 = 16)
Up to (15^2 = 225)

(\sqrt{a} \times \sqrt{b} = \sqrt{ab})
- Example (\sqrt{100} = \sqrt{4} \times \sqrt{25} = 10).

Non-Perfect Square Radicand: If no perfect square factor exists, it remains in simplest form.
Simplifying Process:
- Identify a perfect square factor.
- Simplify the perfect square, and leave remaining factor as radicand.
- Example: (\sqrt{12} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}).

Even Exponents: Always perfect squares.
- Example: (x^2 = x), (x^6 = x^3).
Odd Exponents: Not perfect squares, but can be simplified.
- Subtract 1 from exponent, simplify perfect square factor.
- Example: (x^3 = x^2 \times x = x(x)).

(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}})
- Example: (\frac{\sqrt{100}}{\sqrt{4}} = \sqrt{\frac{100}{4}} = 5).

Process: Multiply numerator and denominator by radical denominator to eliminate radicals from denominator.
- Example: (\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}).

Adding & Subtracting: Combine like square roots.
- Example: (\sqrt{9} + \sqrt{4} = 3 + 2 = 5).
Multiplying & Dividing: Use product and quotient rules.
- Simplify before performing operations.
- Example: (\sqrt{4} \times \sqrt{9} = 6).
Conjugates: To rationalize denominators with two terms, multiply by the conjugate.
- Conjugates result in the difference of squares.