Question

express the following fraction in simplest form, only using positive exponents: \\(\frac{-3(v^3)^{-5}}{6s^{-7}}\\)

Explanation:

Step1: Simplify the coefficient

Simplify the fraction of the coefficients \(\frac{-3}{6}\), which simplifies to \(-\frac{1}{2}\).

Step2: Simplify the power of \(v\)

Using the power - of - a - power rule \((a^{m})^{n}=a^{mn}\), for \((v^{3})^{-5}\), we have \(v^{3\times(-5)} = v^{-15}\).

Step3: Simplify the power of \(s\)

Using the negative exponent rule \(a^{-n}=\frac{1}{a^{n}}\) (or \(\frac{1}{a^{-n}}=a^{n}\)), for \(s^{-7}\) in the denominator, when we move it to the numerator, it becomes \(s^{7}\).

Step4: Combine and simplify with positive exponents

Now, substitute the simplified parts back into the original expression:

$$
LATEXBLOCK0
$$

(We used the rule \(a^{-n}=\frac{1}{a^{n}}\) to convert \(v^{-15}\) to \(\frac{1}{v^{15}}\))

Answer:

\(-\frac{s^{7}}{2v^{15}}\)