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

Let's simplify each expression step by step before ordering them.

1. **Expression 1:** \((-2)^{-8} \cdot (2)^{11}\)

\[
(-2)^{-8} = \frac{1}{(-2)^8} = \frac{1}{256}
\]

Since \((-2)^8 = (2^8) = 256\), we have:
\[
(-2)^{-8} \cdot (2)^{11} = \frac{1}{256} \cdot 2^{11} = \frac{2^{11}}{256}
\]
Now calculating \(2^{11} = 2048\):
\[
(-2)^{-8} \cdot (2)^{11} = \frac{2048}{256} = 8
\]

2. **Expression 2:** \((\frac{1}{2})^3\)

\[
(\frac{1}{2})^3 = \frac{1}{2^3} = \frac{1}{8}
\]

3. **Expression 3:** \(2^6 - \frac{2^4}{2^2}\)

\[
2^6 = 64 \quad \text{and} \quad \frac{2^4}{2^2} = 2^{4-2} = 2^2 = 4
\]
Thus,
\[
2^6 - \frac{2^4}{2^2} = 64 - 4 = 60
\]

4. **Expression 4:** \(\frac{1}{4^{-2}}\)

\[
4^{-2} = \frac{1}{4^2} = \frac{1}{16} \quad \text{so} \quad \frac{1}{4^{-2}} = \frac{1}{\frac{1}{16}} = 16
\]

Now we have the simplified values:
- Expression 1: \(8\)
- Expression 2: \(\frac{1}{8}\)
- Expression 3: \(60\)
- Expression 4: \(16\)

Now let's order these values from least to greatest:

\[
\frac{1}{8} < 8 < 16 < 60
\]

Putting it all together, the order from least to greatest is:
\[
(1/2)^3 < (-2)^{-8} \cdot (2)^{11} < 1/4^{-2} < 2^6 - \frac{2^4}{2^2}
\]

Thus, the final order of the expressions is:
1. \((\frac{1}{2})^3\)
2. \((-2)^{-8} \cdot (2)^{11}\)
3. \(\frac{1}{4^{-2}}\)
4. \(2^6 - \frac{2^4}{2^2}\)