Question

which expression is equivalent to \\(\frac{45m^{-6}p^{2}v^{12}}{15m^{-2}p^{8}v^{-4}}\\) for all values of \\(m\\), \\(p\\), and \\(v\\) where the expression is defined?\
a \\(\frac{3v^{8}}{m^{8}p^{6}}\\)\
b \\(\frac{3v^{16}}{m^{4}p^{6}}\\)\
c \\(\frac{30m^{3}}{p^{4}v^{3}}\\)\
d \\(\frac{30v^{3}}{m^{3}p^{4}}\\)\
\\(\circ\\) b\
\\(\circ\\) d\
\\(\circ\\) c\
\\(\circ\\) a

Explanation:

Step1: Simplify coefficients

$\frac{45}{15} = 3$

Step2: Simplify $m$ terms

$m^{-6 - (-2)} = m^{-6+2} = m^{-4} = \frac{1}{m^4}$

Step3: Simplify $p$ terms

$p^{2 - 8} = p^{-6} = \frac{1}{p^6}$

Step4: Simplify $v$ terms

$v^{12 - (-4)} = v^{12+4} = v^{16}$

Step5: Combine all simplified parts

$3 \cdot \frac{1}{m^4} \cdot \frac{1}{p^6} \cdot v^{16} = \frac{3v^{16}}{m^4p^6}$

Answer:

B. $\frac{3v^{16}}{m^4p^6}$