Recording-2025-07-09T13:57:38.069Z

Review of Basic Exponent Laws (Chapter P refresher)

• Multiplication of like bases: a^x\,\cdot\,a^y=a^{x+y}
  • “Same base add exponents.”
• Negative exponents: a^{-x}=\dfrac{1}{a^x}
• Zero exponent: a^0=1 (NOT 0)
• First power: a^1=a
• Power of a product: (ab)^x=a^x\,b^x
• Power of a quotient: \Bigl(\dfrac{a}{b}\Bigr)^x=\dfrac{a^x}{b^x}
• Square-root fact repeatedly used: \sqrt{a}\,\cdot\,\sqrt{a}=a

Worked Simplification Examples

Example 1: (6^{\sqrt{3}})^{\sqrt{3}}

• Apply power-of-a-power rule: multiply exponents.
  \sqrt{3}\times\sqrt{3}=3 ⇒ expression becomes 6^3=216.

Example 2: 6^{\sqrt{27}}\,\cdot\,6^{\sqrt{12}}

1. Combine exponents (same base):
   6^{\sqrt{27}+\sqrt{12}}
2. Rewrite each radical to expose perfect squares.
   • \sqrt{27}=\sqrt{9\cdot3}=3\sqrt{3}
   • \sqrt{12}=\sqrt{4\cdot3}=2\sqrt{3}
3. Add “like terms”: 3\sqrt{3}+2\sqrt{3}=5\sqrt{3}
4. Final compact form: 6^{5\sqrt{3}}
   (Textbook pursues: 6^5=7,776\;\Rightarrow\;7,776^{\sqrt{3}}; instructor notes this level of arithmetic will NOT be required on exams.)

Graphing the Basic Exponential Function

Prototype: f(x)=2^x

• Point table (chosen to reveal symmetry around the y-axis):
  • x=-2: f=2^{-2}=\tfrac14
  • x=-1: f=\tfrac12
  • x=0: f=1 (universal y-intercept for a^x)
  • x=1: f=2
  • x=2: f=4
• Qualitative behaviour
  • Increasing everywhere (base a=2>1).
  • Domain (-\infty,\infty).
  • Range (0,\infty) – never touches x-axis (horizontal asymptote y=0).
  • No x-intercept.

Base between 0 and 1: f(x)=\bigl(\tfrac12\bigr)^x

• Same $x$-values give: $$1,\tfrac