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:
- \((-2)^{-5} / (-2)^{-10} \rightarrow -32\)
- \(2^{-1} \cdot 2^{4} \rightarrow 8\)
- \(-(-\frac{1}{2})^{3} \cdot (-\frac{1}{2})^{2} \rightarrow \frac{1}{32}\)
- \(\frac{2}{2^{-4}} \rightarrow 32\)
If you have specific expressions you need matched or corrected, please clarify!
3 answers