Question

express the following fraction in simplest form, only using positive exponents.\\(\frac{(2p^{-1})^{5}}{6j^{-8}p^{-8}}\\)

Explanation:

Step1: 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 \((2p^{-1})^5\), we have \(2^5 \times (p^{-1})^5\). Calculating \(2^5 = 32\) and using the power of a power rule \((a^m)^n = a^{mn}\), we get \((p^{-1})^5 = p^{-5}\). So the numerator becomes \(32p^{-5}\).

Step2: Simplify the exponents of \(p\) using the quotient rule for exponents.

The quotient rule for exponents is \(\frac{a^m}{a^n} = a^{m - n}\). For the \(p\) terms, we have \(\frac{p^{-5}}{p^{-8}}\). Using the quotient rule, this is \(p^{-5 - (-8)} = p^{3}\) (because subtracting a negative is adding the positive, so \(-5 + 8 = 3\)).

Step3: Simplify the exponents of \(j\) using the negative exponent rule.

The negative exponent rule states that \(a^{-n} = \frac{1}{a^n}\) and \(\frac{1}{a^{-n}} = a^n\). For the \(j\) term, we have \(\frac{1}{j^{-8}} = j^{8}\) (since \(\frac{1}{j^{-8}} = j^{8}\) by the negative exponent rule).

Step4: Simplify the constant fraction.

Simplify \(\frac{32}{6}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2. So \(\frac{32}{6} = \frac{16}{3}\).

Step5: Combine all the simplified terms.

Putting it all together, we have \(\frac{16}{3} \times j^{8} \times p^{3}\).

Answer:

\(\frac{16j^{8}p^{3}}{3}\)