To simplify the expression \(\left(\frac{P^{-2 / 3} Q^{5 / 6}}{P^{3 / 4} Q^{7 / 2}}\right)\), we will handle the exponents of \(P\) and \(Q\) separately.
### Step 1: Simplify the Exponent of \(P\)
First, let's examine the exponents of \(P\):
- The exponent of \(P\) in the numerator is \(-\frac{2}{3}\).
- The exponent of \(P\) in the denominator is \(\frac{3}{4}\).
When dividing two terms with the same base, we subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
P^{-2/3 - 3/4}
\][/tex]
### Step 2: Simplify the Exponent of \(Q\)
Now, let's examine the exponents of \(Q\):
- The exponent of \(Q\) in the numerator is \(\frac{5}{6}\).
- The exponent of \(Q\) in the denominator is \(\frac{7}{2}\).
Similarly, when dividing two terms with the same base, we subtract the exponent in the denominator from the exponent in the numerator:
[tex]\[
Q^{5/6 - 7/2}
\][/tex]
### Step 3: Perform the Subtraction for Exponents
Let's calculate the actual values of these subtractions individually:
- For the exponent of \(P\):
\(-\frac{2}{3} - \frac{3}{4}\)
- For the exponent of \(Q\):
\(\frac{5}{6} - \frac{7}{2}\)
### Step 4: Resulting Expression
From the results, we have:
[tex]\[P^{-1.41666666666667} \quad \text{and} \quad Q^{-2.66666666666667}\][/tex]
Putting these together, we get:
[tex]\[\frac{P^{-2 / 3} Q^{5 / 6}}{P^{3 / 4} Q^{7 / 2}} = P^{-1.41666666666667} \cdot Q^{-2.66666666666667}\][/tex]
### Step 5: Rewriting the Expression with Positive Exponents
It's often preferable to write expressions with positive exponents. Using the property of exponents \(a^{-b} = \frac{1}{a^b}\), we get:
[tex]\[
P^{-1.41666666666667} \cdot Q^{-2.66666666666667} = \frac{1}{P^{1.41666666666667} \cdot Q^{2.66666666666667}}
\][/tex]
### Final Simplified Expression
Therefore, the expression simplifies to:
[tex]\[
\boxed{\frac{1}{P^{1.41666666666667} Q^{2.66666666666667}}}
\][/tex]
