Question

Need help please worth 50 points

Answer 1

Here's how you simplify those. The rule for raising an exponent to an exponent is that you multiply them. The rule for dividing exponents with the same base is that you subtract the denomonator from the numerator.  So let's simplify the first one, which is actually the most confusing.
$( \frac{ p^{5} }{ p^{-3} q^{-4} } )^{ \frac{1}{4} }$
Multiplying the exponents you get:
$( \frac{ p^{ \frac{5}{4} } }{ p^{- \frac{3}{4} } q^{-1} })$
Subtracting the denominator from the numerator between the common base of p you get this:
$\frac{ p^{ \frac{5}{4}- (-\frac{3}{4}) } }{ q^{-1} }$
Doing that math gives you
$\frac{ p^{2} }{ \frac{1}{q} }$ which equals $\frac{ p^{2} }{1} * \frac{q}{1}$ which is $p^{2}q$, or the third one down.

The next one:
$( \frac{ p^{2} q^{7} }{ q^{4} } )^{ \frac{1}{2} }$
simplifies to:
$\frac{p* q^{ \frac{7}{2} } }{ q^{2} }$
and subtracting the power of the denominator from the power of the numerator gives you:
$p q^{ \frac{3}{2} }$, which is the last choice.
$( p^{6}q \frac{3}{2})^{ \frac{1}{3} }$
simplifies to:
$( p^{6* \frac{1}{3} } )( q^{ \frac{3}{2}* \frac{1}{3} })$
which simplifies very nicely to:
$p^{2}q^{ \frac{1}{2} }$, which is the first choice. The other one is found by process of elimination!