Question

Put the simplified expressions in order on the number line from least to greatest.

1. 1/4^-2
2. 2^6 -2^4/2^2
3. (1/2)^-3
4. (-2)^-8 * (-2)^11

Answer 1

Let's simplify each expression step by step:

1. **\( \frac{1}{4^{-2}} \)**:
\[
4^{-2} = \frac{1}{4^2} = \frac{1}{16}
\]
Therefore:
\[
\frac{1}{4^{-2}} = 4^2 = 16
\]

2. **\( 2^6 - \frac{2^4}{2^2} \)**:
First, simplify \( \frac{2^4}{2^2} \):
\[
\frac{2^4}{2^2} = 2^{4 - 2} = 2^2 = 4
\]
Then, calculate \( 2^6 - 4 \):
\[
2^6 = 64 \quad \text{so} \quad 64 - 4 = 60
\]

3. **\( \left( \frac{1}{2} \right)^{-3} \)**:
\[
\left( \frac{1}{2} \right)^{-3} = 2^3 = 8
\]

4. **\( (-2)^{-8} * (-2)^{11} \)**:
When multiplying powers with the same base, we add the exponents:
\[
(-2)^{-8} * (-2)^{11} = (-2)^{-8 + 11} = (-2)^3
\]
Calculate \( (-2)^3 \):
\[
(-2)^3 = -8
\]

Now we have simplified all expressions:
1. \( 16 \)
2. \( 60 \)
3. \( 8 \)
4. \( -8 \)

Putting these values in order from least to greatest:
- \( -8 \) (from expression 4)
- \( 8 \) (from expression 3)
- \( 16 \) (from expression 1)
- \( 60 \) (from expression 2)

Therefore, the order is:
\[
\text{4, 3, 1, 2}
\]