Question

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

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(4 points)
Arrange responses in the correct order to answer the question. Select a response, navigate to the desired position and insert response at that position. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can be moved using the up and down arrow keys or by dragging with a mouse.
Responses:
1/4^-2
2^6-2^4/2^2
(1/2)^-3
(-2)^-8 * (-2)^11

Answer 1

To compare the expressions, we'll simplify each one step by step.

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

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

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

4. **\( (-2)^{-8} \cdot (-2)^{11} \)**:
Use the exponent rule \( a^m \cdot a^n = a^{m+n} \):
\[
(-2)^{-8} \cdot (-2)^{11} = (-2)^{-8 + 11} = (-2)^{3} = -2^3 = -8
\]

Now we have evaluated all expressions:
- \( \frac{1}{4^{-2}} = 16 \)
- \( 2^6 - \frac{2^4}{2^2} = 60 \)
- \( \left(\frac{1}{2}\right)^{-3} = 8 \)
- \( (-2)^{-8} \cdot (-2)^{11} = -8 \)

Now we can order these from least to greatest:
- \( -8 \)
- \( 8 \)
- \( 60 \)
- \( 16 \)

Thus, the order from least to greatest is:
1. **\((-2)^{-8} \cdot (-2)^{11}\)**
2. **\((\frac{1}{2})^{-3}\)**
3. **\(2^6 - \frac{2^4}{2^2}\)**
4. **\(\frac{1}{4^{-2}}\)**

In summary:
- \( (-8, 8, 60, 16) \).

Arranged correctly on the number line from least to greatest:
- \( (-2)^{-8} \cdot (-2)^{11} \)
- \( \left(\frac{1}{2}\right)^{-3} \)
- \( 2^6 - \frac{2^4}{2^2} \)
- \( \frac{1}{4^{-2}} \)