How do you simplify (3p)^4 (3p^-9)?

Feb 16, 2017

See the entire simplification process below:

Explanation:

First, we can use these rules for exponents to simplify the term on the left of the expression:

$a = {a}^{\textcolor{red}{1}}$ and ${\left({x}^{\textcolor{red}{a}}\right)}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} \times \textcolor{b l u e}{b}}$

${\left({3}^{\textcolor{red}{1}} {p}^{\textcolor{red}{1}}\right)}^{\textcolor{b l u e}{4}} \left(3 {p}^{-} 9\right) = \left({3}^{\textcolor{red}{1} \times \textcolor{b l u e}{4}} {p}^{\textcolor{red}{1} \times \textcolor{b l u e}{4}}\right) \left(3 {p}^{-} 9\right) = \left({3}^{4} {p}^{4}\right) \left(3 {p}^{-} 9\right) =$

$\left(81 {p}^{4}\right) \left(3 {p}^{-} 9\right)$

We can next rewrite this expression as:

$\left(81 \times 3\right) \left({p}^{4} {p}^{-} 9\right) = 243 {p}^{4} {p}^{-} 9$

Then, we can use this rule for exponents to simplify the $p$ terms:

${x}^{\textcolor{red}{a}} \times {x}^{\textcolor{b l u e}{b}} = {x}^{\textcolor{red}{a} + \textcolor{b l u e}{b}}$

$243 {p}^{\textcolor{red}{4}} {p}^{\textcolor{b l u e}{- 9}} = 243 {p}^{\textcolor{red}{4} + \textcolor{b l u e}{- 9}} = 243 {p}^{-} 5$

If we want an expression with no negative exponents we can now use this rule for exponents:

${x}^{\textcolor{red}{a}} = \frac{1}{x} ^ \textcolor{red}{- a}$

$243 {p}^{\textcolor{red}{- 5}} = \frac{243}{p} ^ \textcolor{red}{- - 5} = \frac{243}{p} ^ 5$

The solution to this problem is:

$243 {p}^{-} 5$ or $\frac{243}{p} ^ 5$