Question

How do you simplify ${{\left( \dfrac{{{m}^{3}}{{p}^{5}}}{{{n}^{7}}} \right)}^{6}}.{{\left( \dfrac{{{m}^{2}}{{n}^{0}}{{p}^{3}}}{{{m}^{4}}{{n}^{2}}} \right)}^{3}}$ and write it using only positive exponents ?

Answer 1

Hint: We will simplify the given expression using the properties of exponentiation. We will use following properties of exponentiation:
$\begin{align}
  & {{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}} \\
 & \dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}} \\
 & {{a}^{m}}\times {{a}^{n}}={{a}^{m+n}} \\
\end{align}$

Complete step-by-step answer:
We have been given an expression ${{\left( \dfrac{{{m}^{3}}{{p}^{5}}}{{{n}^{7}}} \right)}^{6}}.{{\left( \dfrac{{{m}^{2}}{{n}^{0}}{{p}^{3}}}{{{m}^{4}}{{n}^{2}}} \right)}^{3}}$.
We have to simplify the given expression and write it using only positive exponents.
To simplify the given expression first we will solve the powers given in the expression.
We know that ${{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}}$.
Now, by applying the above property to the given expression we will get
$\Rightarrow \left( \dfrac{{{m}^{3\times 6}}{{p}^{5\times 6}}}{{{n}^{7\times 6}}} \right).{{\left( \dfrac{{{m}^{2\times 3}}{{n}^{0\times 3}}{{p}^{3\times 3}}}{{{m}^{4\times 3}}{{n}^{2\times 3}}} \right)}^{3}}$
Now, simplifying the above obtained equation we will get
$\Rightarrow \left( \dfrac{{{m}^{18}}{{p}^{30}}}{{{n}^{42}}} \right).\left( \dfrac{{{m}^{6}}{{n}^{0}}{{p}^{9}}}{{{m}^{12}}{{n}^{6}}} \right)$
Now, we know that ${{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}$
Now, by applying the above property to the given expression we will get
$\Rightarrow \left( \dfrac{{{m}^{18+6}}{{p}^{30+9}}}{{{m}^{12}}{{n}^{42+6}}} \right)$
Now, simplifying the above obtained equation we will get
$\Rightarrow \left( \dfrac{{{m}^{24}}{{p}^{39}}}{{{m}^{12}}{{n}^{48}}} \right)$
Now, we know that $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$
Now, by applying the above property to the given expression we will get
$\Rightarrow \left( \dfrac{{{m}^{24-12}}{{p}^{39}}}{{{n}^{48}}} \right)$
Now, simplifying the above obtained equation we will get
$\Rightarrow \left( \dfrac{{{m}^{12}}{{p}^{39}}}{{{n}^{48}}} \right)$
Hence above is the required simplified form of given expression with positive exponents.

Note: The point to be noted is that here in this particular question we have to express the variables with positive exponents only. If we get a negative exponent we can convert it into a positive exponent as ${{a}^{-n}}=\dfrac{1}{{{a}^{n}}}$ . To solve such types of questions we must have knowledge of properties of exponentiation.