Question

Put the simplified expressions in order on the number line from least to greatest.
(-2)^-8.(2)^11

(1/2)^-3

2^6 -2^4/2^2

1/4^-2

Answer 1

To simplify the expressions given, let's evaluate each one step by step.

1. **For \((-2)^{-8} \cdot (2)^{11}\)**:
\[
(-2)^{-8} = \frac{1}{(-2)^8} = \frac{1}{256}
\]
\[
(2)^{11} = 2048
\]
Thus,
\[
(-2)^{-8} \cdot (2)^{11} = \frac{1}{256} \cdot 2048 = \frac{2048}{256} = 8
\]

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

3. **For \(2^6 - \frac{2^4}{2^2}\)**:
\[
2^6 = 64,
\]
\[
\frac{2^4}{2^2} = 2^{4-2} = 2^2 = 4
\]
Thus,
\[
2^6 - \frac{2^4}{2^2} = 64 - 4 = 60
\]

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

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

Putting these values in order from least to greatest:
- **60**
- **16**
- **8**
- **8**

So, if we order everything from least to greatest:
1. \(60\)
2. \(16\)
3. \(8\)
4. \(8\)

The final order from least to greatest is:
- \(60\)
- \(16\)
- \(8\)
- \(8\) (but both are the same, so they can be grouped)

Thus, the final presentation would be:
1. \(60\)
2. \(16\)
3. \(8\) (doubled)

If you need them uniquely, \(60 < 16 < 8\).