Question

How do you simplify $\dfrac{{12{m^{ - 4}}{p^2}}}{{ - 15{m^3}{p^{ - 9}}}}$ ?

Answer 1

Hint: To solve the question, first we will rewrite the expression and try to break it in its very simplest form. We will also use the rule for exponents in these given expressions. We will break the expression until it achieves its non-operational state.

Complete step-by-step solution:
First we rewrite the expression as:
$
\Rightarrow (\dfrac{{12}}{{ - 15}})(\dfrac{{{m^{ - 4}}}}{{{m^3}}})(\dfrac{{{p^2}}}{{{p^{ - 9}}}}) \\
\Rightarrow - \dfrac{{3 \times 4}}{{3 \times 5}}(\dfrac{{{m^{ - 4}}}}{{{m^3}}})(\dfrac{{{p^2}}}{{{p^{ - 9}}}}) \\
 \Rightarrow - \dfrac{4}{5}(\dfrac{{{m^{ - 4}}}}{{{m^3}}})(\dfrac{{{p^2}}}{{{p^{ - 9}}}}) \\
$
Next, use this rule for exponents to simplify the $m$ terms:
$\dfrac{{{x^a}}}{{{x^b}}} = \dfrac{1}{{{x^{b - a}}}}$
$
  \because - \dfrac{4}{5}(\dfrac{{{m^{ - 4}}}}{{{m^3}}})(\dfrac{{{p^2}}}{{{p^{ - 9}}}}) \\
  \Rightarrow - \dfrac{4}{5}(\dfrac{1}{{{m^{3 - - 4}}}})(\dfrac{{{p^2}}}{{{p^{ - 9}}}}) \\
   \Rightarrow - \dfrac{4}{5}(\dfrac{1}{{{m^{3 + 4}}}})(\dfrac{{{p^2}}}{{{p^{ - 9}}}}) \\
  \Rightarrow - \dfrac{4}{5}(\dfrac{1}{{{m^7}}})(\dfrac{{{p^2}}}{{{p^{ - 9}}}}) \\
   \Rightarrow - \dfrac{4}{{5{m^7}}}(\dfrac{{{p^2}}}{{{p^{ - 9}}}}) \\
$
Now, use this rule for exponents to simplify the $p$ terms:
$\dfrac{{{x^a}}}{{{x^b}}} = {x^{a - b}}$
$
  \because - \dfrac{4}{{5{m^7}}}(\dfrac{{{p^2}}}{{{p^{ - 9}}}}) \\
   \Rightarrow - \dfrac{4}{{5{m^7}}}{p^{2 - - 9}} \\
   \Rightarrow - \dfrac{4}{{5{m^7}}}{p^{2 + 9}} \\
   \Rightarrow - \dfrac{4}{{5{m^7}}}{p^{11}} \\
   \Rightarrow - \dfrac{{4{p^{11}}}}{{5{m^7}}} \\
$

Hence, the simplification of $\dfrac{{12{m^{ - 4}}{p^2}}}{{ - 15{m^3}{p^{ - 9}}}}$ is $ - \dfrac{{4{p^{11}}}}{{5{m^7}}}$ .

Note: This leads to another rule for exponents—the Power Rule for Exponents. To simplify a power of a power, you multiply the exponents, keeping the base the same.