Question

Powers of 3 - Value
3^(3) - 27
3^(2) - 9
3^(1) - 3
3^(0) - a
3^(-1) - b
3^(-2) - 1/9
3^(-3) - c

What is the patteran as the exponents decrease? _
(divide by 2, divide by 9, subtract by 6, divide by 3)

What is the value of each variable?
a = _
b = _
c = _

Answer 1

To find the values of \( a \), \( b \), and \( c \) based on the powers of 3 and the pattern of division as the exponents decrease, let's look at the powers step by step.

1. **Starting from \( 3^3 = 27 \)**, we can calculate downwards:
- \( 3^2 = \frac{3^3}{3} = \frac{27}{3} = 9 \)
- \( 3^1 = \frac{3^2}{3} = \frac{9}{3} = 3 \)
- \( 3^0 = \frac{3^1}{3} = \frac{3}{3} = 1 \)
- Thus, \( a = 1 \)

2. **Continuing to negative exponents**:
- \( 3^{-1} = \frac{3^0}{3} = \frac{1}{3} \)
- Thus, \( b = \frac{1}{3} \)

3. **Next, for \( 3^{-2} \)**:
- \( 3^{-2} = \frac{3^{-1}}{3} = \frac{\frac{1}{3}}{3} = \frac{1}{9} \)
- The value \( 3^{-2} \) is already given as \( \frac{1}{9} \), thus confirmed.

4. **Finally, for \( 3^{-3} \)**:
- \( 3^{-3} = \frac{3^{-2}}{3} = \frac{\frac{1}{9}}{3} = \frac{1}{27} \)
- Thus, \( c = \frac{1}{27} \)

So, the values are:
- \( a = 1 \)
- \( b = \frac{1}{3} \)
- \( c = \frac{1}{27} \)

Additionally, regarding the pattern as the exponents decrease:
- Moving from \( 3^3 \) to \( 3^2 \) involves dividing by 3,
- Moving from \( 3^2 \) to \( 3^1 \) also involves dividing by 3,
- Moving from \( 3^1 \) to \( 3^0 \) also involves dividing by 3,
- Moving from \( 3^0 \) to \( 3^{-1} \), \( 3^{-1} \) to \( 3^{-2} \), and so forth also involves dividing by 3.

Thus, the complete answer is:
- **The pattern when the exponents decrease is to divide by 3.**
- \( a = 1 \)
- \( b = \frac{1}{3} \)
- \( c = \frac{1}{27} \)