Question

How do you simplify the expression \[\dfrac{{{r}^{-3}}{{s}^{5}}{{t}^{2}}}{{{r}^{2}}s{{t}^{-2}}}\]?

Answer 1

Hint: Now we know that $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$ hence using this law we will remove fraction from the given expression. Now we will simplify the expression. Now we know that ${{a}^{0}}=1$ and ${{a}^{-n}}=\dfrac{1}{{{a}^{n}}}$ hence using this we will easily simplify the given expression.

Complete step by step solution:
First let us understand the concept of indices.
Now index or indices are nothing but a number or variable raised to power.
Now for example consider the number ${{4}^{3}}$ .
Here 3 is the power of 4. The number is read as 3 raised to 4.
Now let us understand the meaning of power of a number.
The power tells us how many times a number is multiplied by itself.
Hence ${{4}^{3}}$ means 4 multiplied by itself 3 times.
Hence we have,
 $\Rightarrow {{4}^{3}}=4\times 4\times 4$
Now note that if the power of a number or variable is 0 then the value of the expression is 1.
Hence we get ${{a}^{0}}=1$
Also we can convert the negative power into positive by simply putting the term in denominator hence we have, ${{a}^{-n}}=\dfrac{1}{{{a}^{n}}}$
Now let us learn some laws of indices. There are three basic properties of indices.
$\begin{align}
  & 1){{a}^{m}}{{a}^{n}}={{a}^{m+n}} \\
 & 2)\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}} \\
 & 3){{\left( {{a}^{m}} \right)}^{n}}={{a}^{mn}} \\
\end{align}$
Now consider the given expression \[\dfrac{{{r}^{-3}}{{s}^{5}}{{t}^{2}}}{{{r}^{2}}s{{t}^{-2}}}\]
Now let us club same variables together. Hence we get,
$\Rightarrow \left( \dfrac{{{r}^{-3}}}{{{r}^{2}}} \right)\left( \dfrac{{{s}^{5}}}{s} \right)\left( \dfrac{{{t}^{2}}}{{{t}^{-2}}} \right)$
Now first using the property $\dfrac{{{a}^{m}}}{{{a}^{n}}}={{a}^{m-n}}$ we get,
$\Rightarrow \left( {{r}^{-3-2}} \right)\left( {{s}^{5-1}} \right)\left( {{t}^{2+2}} \right)$
Now on simplifying we get,
$\begin{align}
  & \Rightarrow {{r}^{-5}}{{s}^{4}}{{t}^{4}} \\
 & \Rightarrow {{r}^{-5}}{{s}^{4}} {{t}^{4}} \\
 & \Rightarrow \dfrac{{{t}^{4}} {{s}^{4}}}{{{r}^{5}}} \\
\end{align}$

Hence the simplified expression is $\dfrac{{{t}^{4}} {{s}^{4}}}{{{r}^{5}}}$

Note: Now note that for each law of indices we have the variable of number to be the same. Hence we have no rule for ${{a}^{m}}$ and ${{b}^{n}}$ . We can prove all the three laws by simply expanding the index into multiplication.