The problem asks for the evaluation of the exponential function $f(x) = 6^x$ at specific values of $x$. Let's solve this step by step.

Given:

• $f(x) = 6^x$
• Find:
  • $\text{Base} =$ ?
  • $f(-2) =$ ?
  • $f(0) =$ ?
  • $f(2) =$ ?
  • $f(6) =$ ?

Step-by-step solution:

1. Base of the exponential function:

   • The base of $f(x) = 6^x$ is 6.

2. Evaluate $f(-2)$: $f(-2) = 6^{-2} = \frac{1}{6^2} = \frac{1}{36}.$

3. Evaluate $f(0)$: $f(0) = 6^0 = 1 \quad \text{(any number raised to the power of 0 equals 1)}.$

4. Evaluate $f(2)$: $f(2) = 6^2 = 36.$

5. Evaluate $f(6)$: $f(6) = 6^6 = 46656 \quad \text{(calculated as } 6 \times 6 \times 6 \times 6 \times 6 \times 6\text{)}.$

Final Answers:

• Base = $6$
• $f(-2) = \frac{1}{36}$
• $f(0) = 1$
• $f(2) = 36$
• $f(6) = 46656$

Would you like me to explain any step in more detail?

Related Questions:

1. What is the general form of an exponential function?
2. How is $6^{-2}$ derived as a fraction?
3. What is the significance of the base in an exponential function?
4. How do you evaluate $6^6$ without a calculator?
5. Why is any number raised to the power of 0 equal to 1?

Tip:

When working with exponential functions, always remember the key property: $a^{-n} = \frac{1}{a^n}$.