Question

express the following fraction in simplest form, only using positive exponents.\\(dfrac{-3(h^{3}t^{-5})^{-3}}{6h^{4}}\\)

Explanation:

Step1: Simplify the numerator's exponent

Using the power of a product rule \((ab)^n = a^n b^n\) and the power of a power rule \((a^m)^n = a^{mn}\), we have:
\((h^{3}t^{-5})^{-3}=h^{3\times(-3)}t^{-5\times(-3)} = h^{-9}t^{15}\)
So the numerator becomes \(-3h^{-9}t^{15}\)

Step2: Simplify the fraction's coefficient and exponents

First, simplify the coefficient: \(\frac{-3}{6}=-\frac{1}{2}\)
Then, for the \(h\) terms, use the quotient rule \(a^m\div a^n=a^{m - n}\): \(h^{-9}\div h^{4}=h^{-9 - 4}=h^{-13}\)
The \(t\) term remains \(t^{15}\) as there is no \(t\) in the denominator.

Step3: Convert negative exponents to positive

Using the rule \(a^{-n}=\frac{1}{a^{n}}\), \(h^{-13}=\frac{1}{h^{13}}\)
So putting it all together: \(-\frac{1}{2}\times\frac{t^{15}}{h^{13}}=-\frac{t^{15}}{2h^{13}}\)

Answer:

\(-\frac{t^{15}}{2h^{13}}\)