Sure, let's carefully analyze and match each expression with its equivalent using the given numerical results and step-by-step logic.
### Given Expressions
1. [tex]\(-\frac{1}{125}\)[/tex]
2. [tex]\(1\)[/tex]
3. [tex]\(-1\)[/tex]
4. [tex]\(-125\)[/tex]
5. [tex]\(\frac{1}{125}\)[/tex]
### Exponential Expressions to be Matched
- [tex]\(5^{-3}\)[/tex]
- [tex]\(-5^{-3}\)[/tex]
- [tex]\(\left(-5^{-3}\right)^{-1}\)[/tex]
- [tex]\(\left(-5^{-3}\right)^0\)[/tex]
### Matching Each Expression
#### 1. [tex]\(5^{-3}\)[/tex]
- We know from the rule of exponents [tex]\(a^{-n} = \frac{1}{a^n}\)[/tex] that [tex]\(5^{-3} = \frac{1}{5^3} = \frac{1}{125}\)[/tex]
- Comparing this with the numerical result, [tex]\(\frac{1}{125}\)[/tex] matches [tex]\(5^{-3}\)[/tex]
#### 2. [tex]\(-5^{-3}\)[/tex]
- Here, we introduce a negative sign in front of [tex]\(5^{-3}\)[/tex]
- Therefore, [tex]\(-5^{-3} = -\left(\frac{1}{5^3}\right) = -\frac{1}{125}\)[/tex]
- Comparing this with the numerical result, [tex]\(-\frac{1}{125}\)[/tex] matches [tex]\(-5^{-3}\)[/tex]
#### 3. [tex]\(\left(-5^{-3}\right)^{-1}\)[/tex]
- Exponent rule tells us [tex]\((a^m)^n = a^{mn}\)[/tex], so [tex]\(\left(-5^{-3}\right)^{-1} = (-5)^{3 \times -1} = (-5)^3\)[/tex]
- Calculate [tex]\((-5)^3\)[/tex]:
[tex]\[
(-5)^3 = -125
\][/tex]
- Notice the rule of inverse because [tex]\(\left(-a^{-1}\right)^{-1} = a\)[/tex], thus apply the inverse logic to the compound fraction:
So, [tex]\(\left(-5^{-3}\right)^{-1} = -125\)[/tex]
- Comparing this with the numerical result, [tex]\(-125\)[/tex] matches [tex]\(\left(-5^{-3}\right)^{-1}\)[/tex]
#### 4. [tex]\(\left(-5^{-3}\right)^0\)[/tex]
- Any non-zero number raised to the power of 0 is 1.
- Therefore, [tex]\(\left(-5^{-3}\right)^0 = 1\)[/tex]
- Comparing this with the numerical result, [tex]\(1\)[/tex] matches [tex]\(\left(-5^{-3}\right)^0\)[/tex]
### Final Match
- [tex]\(\frac{1}{125} \rightarrow 5^{-3}\)[/tex]
- [tex]\(-\frac{1}{125} \rightarrow -5^{-3}\)[/tex]
- [tex]\(-125 \rightarrow \left(-5^{-3}\right)^{-1}\)[/tex]
- [tex]\(1 \rightarrow \left(-5^{-3}\right)^0\)[/tex]
Therefore, the detailed matching is as follows:
- [tex]\(5^{-3}\)[/tex] matches [tex]\( \frac{1}{125} \)[/tex]
- [tex]\(-5^{-3}\)[/tex] matches [tex]\( -\frac{1}{125} \)[/tex]
- [tex]\(\left(-5^{-3}\right)^{-1}\)[/tex] matches [tex]\(-125\)[/tex]
- [tex]\(\left(-5^{-3}\right)^0\)[/tex] matches [tex]\(1\)[/tex]
So, the expressions can be matched as follows:
- [tex]\( -\dfrac{1}{125} \)[/tex] matches with [tex]\(-5^{-3}\)[/tex]
- [tex]\(1\)[/tex] matches with [tex]\(\left(-5^{-3}\right)^0\)[/tex]
- [tex]\(-1\)[/tex] does not seem to be listed
- [tex]\(-125\)[/tex] matches with [tex]\(\left(-5^{-3}\right)^{-1}\)[/tex]
- [tex]\(\frac{1}{125}\)[/tex] matches with [tex]\(5^{-3}\)[/tex]
