Question

express the following fraction in simplest form, only using positive exponents.\\(\frac{(3d^{-4}n^{5})^{-3}}{12d^{3}n^{4}}\\)

Explanation:

Step 1: Apply the power of a product rule to the numerator

The power of a product rule states that \((ab)^n = a^n b^n\). So for \((3d^{-4}n^{5})^{-3}\), we have:
\(3^{-3}(d^{-4})^{-3}(n^{5})^{-3}\)
\(=\frac{1}{3^{3}}d^{(-4)\times(-3)}n^{5\times(-3)}\)
\(=\frac{1}{27}d^{12}n^{-15}\)
So the numerator becomes \(\frac{d^{12}n^{-15}}{27}\)

Step 2: Rewrite the fraction with the new numerator

Now our expression is \(\frac{\frac{d^{12}n^{-15}}{27}}{12d^{3}n^{4}}\)
Dividing by a fraction is the same as multiplying by its reciprocal, so this is \(\frac{d^{12}n^{-15}}{27}\times\frac{1}{12d^{3}n^{4}}\)
\(=\frac{d^{12}n^{-15}}{27\times12d^{3}n^{4}}\)
\(=\frac{d^{12}n^{-15}}{324d^{3}n^{4}}\)

Step 3: Use the quotient rule for exponents (\(a^m\div a^n=a^{m - n}\))

For the \(d\) terms: \(d^{12}\div d^{3}=d^{12 - 3}=d^{9}\)
For the \(n\) terms: \(n^{-15}\div n^{4}=n^{-15 - 4}=n^{-19}=\frac{1}{n^{19}}\)
For the constants: \(\frac{1}{324}\)
Putting it all together: \(\frac{d^{9}}{324n^{19}}\)

Answer:

\(\frac{d^{9}}{324n^{19}}\)