To simplify the given mathematical expression, we need to combine terms in the numerator and the denominator appropriately, following the laws of exponents. Here is the step-by-step process:
### Step 1: Rewrite both the numerator and the denominator
Given:
[tex]\[
\frac{\frac{a^{-4} b^{-2} c d^6 e^7}{c d^{[?]} e}}{\frac{a b}{1}}
\][/tex]
### Step 2: Simplify the Powers
We need to rewrite all terms with their respective exponents:
- Numerator: \( a^{-4} b^{-2} c d^6 e^7 \)
- Denominator: \( a b c d^{[?]} e \)
### Step 3: Combine Like Terms in the Denominator and Numerator
Align the like terms (same base):
- For the base \(a\):
[tex]\[
a^{-4} \quad \text{(numerator)}, \; a^1 \quad \text{(denominator)}
\][/tex]
- For the base \(b\):
[tex]\[
b^{-2} \quad \text{(numerator)}, \; b^1 \quad \text{(denominator)}
\][/tex]
- For the base \(c\):
[tex]\[
c^1 \quad \text{(numerator)}, c^1 \quad \text{(denominator)}
\][/tex]
- For the base \(d\):
[tex]\[
d^6 \quad \text{(numerator)}, \; d^{[?]} \quad \text{(denominator)}
\][/tex]
- For the base \(e\):
[tex]\[
e^7 \quad \text{(numerator)}, \; e^1 \quad \text{(denominator)}
\][/tex]
### Step 4: Subtract Exponents in the Denominator From Exponents in the Numerator
The properties of exponents state that:
[tex]\[
\frac{x^m}{x^n} = x^{m-n}
\][/tex]
- For \(a\): \((-4) - (1) = -5\)
- For \(b\): \((-2) - (-1) = -1 \)
- For \(c\): \((1) - (1) = 0 \quad \text{(c cancels out)}\)
- For \(d\): \((6) - ([?]) = 6\)
- For \(e\): \((7) - (1) = 6\)
### Step 5: Simplified Expression
Substituting these simplified exponents back in, we get:
[tex]\[
a^{-5} b^{-1} d^6 e^6
\][/tex]
### Step 6: Determine the Unknown Exponent \([?]\) for \(d\)
We need to ensure that the \(d\) terms in the final simplified expression are factored out correctly. After simplifying, \(d\) in the denominator should match the exponent after simplification. Thus:
Since there is no \(d\) exponent given in the lower part (implied that it needs to remain constant as \(d\)), after simplification, we find:
[tex]\[
d^{[?]} = 6
\][/tex]
Therefore, the unknown exponent [tex]\([?]\)[/tex] is [tex]\(6\)[/tex].
