To simplify the expression [tex]\( b^0 c^0 d^{-1} \cdot b^{-1} c^{-1} d \)[/tex], we'll follow a systematic approach by applying the fundamental rules of exponents. Let's go through the process step-by-step:
1. Simplifying Constants with Zero Exponents:
- Any non-zero number or variable raised to the power of zero is equal to 1.
- Hence, [tex]\( b^0 = 1 \)[/tex] and [tex]\( c^0 = 1 \)[/tex].
So, the expression becomes:
[tex]\[
1 \cdot 1 \cdot d^{-1} \cdot b^{-1} \cdot c^{-1} \cdot d
\][/tex]
2. Simplifying the Products of Powers of [tex]\( d \)[/tex]:
- Next, multiply the [tex]\( d \)[/tex] terms using the rule [tex]\( a^m \cdot a^n = a^{m+n} \)[/tex].
- Here, we have [tex]\( d^{-1} \cdot d \)[/tex].
Simplifying the multiplication:
[tex]\[
d^{-1} \cdot d = d^{-1 + 1} = d^0
\][/tex]
3. Simplifying [tex]\( d^0 \)[/tex]:
- Since any number or variable raised to the power of zero is 1,
[tex]\[
d^0 = 1
\][/tex]
4. Combining All Simplified Parts:
- We now have [tex]\( 1 \cdot b^{-1} \cdot c^{-1} \cdot 1 \)[/tex],
- Which simplifies simply to [tex]\( b^{-1} \cdot c^{-1} \)[/tex].
5. Expressing with Positive Exponents:
- Recall that [tex]\( b^{-1} = \frac{1}{b} \)[/tex] and [tex]\( c^{-1} = \frac{1}{c} \)[/tex].
- Thus, [tex]\( b^{-1} \cdot c^{-1} = \frac{1}{b} \cdot \frac{1}{c} = \frac{1}{bc} \)[/tex].
So the simplified expression, expressed as a single term without a denominator, is:
[tex]\[
\frac{1}{bc}
\][/tex]
Thus, the simplified form of [tex]\( b^0 c^0 d^{-1} \cdot b^{-1} c^{-1} d \)[/tex] is:
[tex]\[
\frac{1}{bc}
\][/tex]
