Question

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

(-2) to the negative eighth power times (-2) to the eleventh power
(1/2) to the negative third power
1 divided by 4 to the negative second power
2 to the sixth power - 2 to the fourth power divided by 2 to the second power

Answer 1

To order the given expressions from least to greatest, we first need to simplify each expression.

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

Using the property of exponents \(a^m \cdot a^n = a^{m+n}\):

\[
(-2)^{-8} \cdot (-2)^{11} = (-2)^{-8 + 11} = (-2)^{3}
\]

Now calculating \((-2)^{3}\):

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

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

Using the property \(a^{-n} = \frac{1}{a^n}\):

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

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

This can be rewritten using the reciprocal of \(4^{-2}\):

\[
1 / 4^{-2} = 4^{2} = 16
\]

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

First calculate \(2^6\), \(2^4\), and \(2^2\):

\[
2^6 = 64
\]
\[
2^4 = 16
\]
\[
2^2 = 4
\]

Now substituting back:

\[
\frac{2^4}{2^2} = \frac{16}{4} = 4
\]

Now substitute back into the expression:

\[
2^6 - \frac{2^4}{2^2} = 64 - 4 = 60
\]

Now we have simplified the expressions:

- \((-2)^{-8} \cdot (-2)^{11} = -8\)
- \((\frac{1}{2})^{-3} = 8\)
- \(1 / 4^{-2} = 16\)
- \(2^6 - 2^4 / 2^2 = 60\)

Next, we will list these results:

1. \(-8\)
2. \(8\)
3. \(16\)
4. \(60\)

Finally, we can order these on the number line from least to greatest:

\[
\boxed{-8, 8, 16, 60}
\]