Question

Use the rules of exponents to simplify the expressions. Match the expression with its equivalent value.

1. (-2)exponent-5/ (-2)exponent-10

2. 2 exponent-1 ∙ 2 exponent-4
-(-1/2)exponent3 * (-1/2)exponent2
2/ 2exponent-4
3. ∙
1/32
4.
-1/32

Answer 1

To simplify the given expressions using the rules of exponents, let's analyze each one:

### 1. **Expression:** \((-2)^{-5} / (-2)^{-10}\)

Using the rule \(\frac{a^{-m}}{a^{-n}} = a^{n - m}\):

\[
(-2)^{-5} / (-2)^{-10} = (-2)^{-5 - (-10)} = (-2)^{-5 + 10} = (-2)^{5}
\]

Since \((-2)^{5} = -32\), the value is:

\[
-32
\]

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

Using the rule \(a^{m} \cdot a^{n} = a^{m+n}\):

\[
2^{-1} \cdot 2^{4} = 2^{-1 + 4} = 2^{3}
\]

Since \(2^{3} = 8\), the value is:

\[
8
\]

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

Using the rule \(a^{m} \cdot a^{n} = a^{m+n}\):

\[
-(-\frac{1}{2})^{3} \cdot (-\frac{1}{2})^{2} = -(-\frac{1}{2})^{3 + 2} = -(-\frac{1}{2})^{5}
\]

Calculating \((-1/2)^{5}\):

\[
(-\frac{1}{2})^{5} = -\frac{1}{32}
\]
So:

\[
-(-\frac{1}{32}) = \frac{1}{32}
\]

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

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

\[
\frac{2}{2^{-4}} = 2^{1 - (-4)} = 2^{1 + 4} = 2^{5}
\]

Since \(2^{5} = 32\), the value is:

\[
32
\]

### Final Matches:
- \((-2)^{-5} / (-2)^{-10} \rightarrow -32\)
- \(2^{-1} \cdot 2^{4} \rightarrow 8\)
- \(-(-\frac{1}{2})^{3} \cdot (-\frac{1}{2})^{2} \rightarrow \frac{1}{32}\)
- \(\frac{2}{2^{-4}} \rightarrow 32\)

Thus, matching up:
1. \((-2)^{-5} / (-2)^{-10} \rightarrow -32\)
2. \(2^{-1} \cdot 2^{4} \rightarrow 8\)
3. \(-(-\frac{1}{2})^{3} \cdot (-\frac{1}{2})^{2} \rightarrow \frac{1}{32}\)
4. \(\frac{2}{2^{-4}} \rightarrow 32\)

If you have specific expressions you need matched or corrected, please clarify!