To simplify the expression [tex]\(\frac{(p q r)^{-2} r^{\frac{1}{3}}}{(p^2 r)^{-1} q^3}\)[/tex] and write it in the form [tex]\(p^a q^b r^c\)[/tex], we can proceed step-by-step by first expanding and then simplifying the exponents.
1. Expand the expression inside the numerator and the denominator:
The numerator is:
[tex]\[(pqr)^{-2} \cdot r^{1/3}\][/tex]
The denominator is:
[tex]\[(p^2 r)^{-1} \cdot q^3\][/tex]
2. Distribute the exponents inside the numerator and the denominator:
In the numerator, we have:
[tex]\[(pqr)^{-2} \cdot r^{1/3} = p^{-2} \cdot q^{-2} \cdot r^{-2} \cdot r^{1/3}\][/tex]
In the denominator, we have:
[tex]\[(p^2 r)^{-1} \cdot q^3 = (p^2)^{-1} \cdot r^{-1} \cdot q^3 = p^{-2} \cdot r^{-1} \cdot q^3\][/tex]
3. Combine and subtract exponents:
Combine the exponents for [tex]\(p\)[/tex], [tex]\(q\)[/tex], and [tex]\(r\)[/tex] from the numerator and the denominator:
For [tex]\(p\)[/tex]:
[tex]\[
p^{-2} / p^{-2} = p^{-2 - (-2)} = p^{-2 + 2} = p^0
\][/tex]
For [tex]\(q\)[/tex]:
[tex]\[
q^{-2} / q^3 = q^{-2 - 3} = q^{-5}
\][/tex]
For [tex]\(r\)[/tex]:
[tex]\[
r^{-2} \cdot r^{1/3} / r^{-1} = r^{-2 + 1/3 - (-1)} = r^{-2 + 1/3 + 1} = r^{-2 + 1 + 1/3} = r^{-2 + 1.333}
\][/tex]
Simplifying [tex]\(r\)[/tex]:
[tex]\[
r^{-2 + 1.333} = r^{-2 + 4/3} = r^{-6/3 + 4/3} = r^{-2/3}
\][/tex]
4. Write the final simplified expression:
After simplifying the exponents, the expression [tex]\(\frac{(p q r)^{-2} r^{\frac{1}{3}}}{(p^2 r)^{-1} q^3}\)[/tex] can be rewritten in the form [tex]\(p^a q^b r^c\)[/tex] as:
[tex]\[
p^0 \cdot q^{-5} \cdot r^{0.333}
\][/tex]
Based on our simplification, the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] are:
[tex]\[
a = 0, \quad b = -5, \quad c = 0.333
\][/tex]
Therefore, the expression in the form [tex]\(p^a q^b r^c\)[/tex] is:
[tex]\[
p^0 q^{-5} r^{0.333}
\][/tex]
