Question

simplify.
\\(\frac{6m^{9}n^{-3}p^{-1}}{2m^{-7}p^{4}}\\)
write your answer using only positive exponents.

Explanation:

Step1: Simplify the coefficient

Divide the coefficient 6 by 2: $\frac{6}{2} = 3$

Step2: Simplify the \( m \)-terms

Use the rule \( \frac{a^m}{a^n} = a^{m - n} \) for \( m \)-terms: \( \frac{m^9}{m^{-7}} = m^{9 - (-7)} = m^{16} \)

Step3: Simplify the \( n \)-term

The \( n \)-term is \( n^{-3} \), and since there's no \( n \) in the denominator, it remains \( n^{-3} \). To make the exponent positive, we can write it as \( \frac{1}{n^3} \), but in the final expression, we'll handle it appropriately. Wait, actually, since there's no \( n \) in the denominator's non - \( m \) and non - \( p \) terms, the \( n^{-3} \) stays as is for now and we'll convert it to positive exponent later.

Step4: Simplify the \( p \)-terms

Use the rule \( \frac{a^m}{a^n}=a^{m - n} \) for \( p \)-terms: \( \frac{p^{-1}}{p^4}=p^{-1 - 4}=p^{-5} \). To convert the negative exponent to positive, we use the rule \( a^{-n}=\frac{1}{a^n} \), so \( p^{-5}=\frac{1}{p^5} \)

Step5: Combine all terms

Now, combine the coefficient, \( m \)-term, \( n \)-term, and \( p \)-term. We have \( 3\times m^{16}\times n^{-3}\times\frac{1}{p^5} \). Converting \( n^{-3} \) to positive exponent (using \( a^{-n}=\frac{1}{a^n} \)), we get \( \frac{3m^{16}}{n^3p^5} \)

Answer:

\( \frac{3m^{16}}{n^{3}p^{5}} \)