Overview
This lecture covers key algebra skills focusing on the laws of exponents (including integer and rational exponents) and the manipulation of expressions in radical form, with detailed examples of each.
Laws of Exponents
- When multiplying like bases, add their exponents: ( x^a \times x^b = x^{a+b} ).
- When dividing like bases, subtract exponents: ( x^a / x^b = x^{a-b} ).
- Negative exponents indicate reciprocals: ( x^{-a} = 1/x^a ).
- Raising a power to a power, multiply exponents: ( (x^a)^b = x^{a \times b} ).
- Any nonzero number raised to the 0 power equals 1: ( x^0 = 1 ).
Radical Form and Exponent Conversion
- A radical has three parts: the radicand (inside), the index/root (outside), and the radical sign.
- The nth root of ( x^b ), written ( \sqrt[a]{x^b} ), converts to exponent form as ( x^{b/a} ).
- The "power is in the flower" mnemonic: numerator is power, denominator is the root.
Laws of Radicals
- Product law: ( \sqrt{x} \times \sqrt{y} = \sqrt{xy} ).
- Quotient law: ( \sqrt{x} / \sqrt{y} = \sqrt{x/y} ).
- Only combine radicals with the same root and radicand: ( a\sqrt{b} + c\sqrt{b} = (a+c)\sqrt{b} ).
Example Problems
- ( 2x^{5/2} \times -3x^{1/4} = -6x^{11/4} ).
- ( \frac{7x^2y^5}{14xy^{-12}} = \frac{1}{2}xy^{17} ).
- ( (3x^2y)^{-3} = \frac{1}{27x^6y^3} ).
- ( \sqrt{16x^4y^3} = 4x^2y^{3/2} ).
- Simplify ( \sqrt{40} = 2\sqrt{10} ).
- ( \sqrt{8} + \sqrt{32} = 6\sqrt{2} ).
- ( \frac{\sqrt{28}}{\sqrt{7}} = 2 ).
Key Terms & Definitions
- Exponent — the power to which a base is raised.
- Radicand — the number inside a radical symbol.
- Root (Index) — the small number outside the radical indicating the degree of the root.
- Radical Form — an expression containing a root.
- Rational Exponent — an exponent expressed as a fraction, where numerator is the power and denominator is the root.
Action Items / Next Steps
- Practice converting between radical form and exponent form.
- Create and simplify your own problems using the laws of exponents and radicals.