Question

How do you simplify the following and write answers with positive exponents only \[-3{{a}^{2}}{{b}^{-5}}\]?

Answer 1

Hint: Assume the given expression as E. Now, leave the numerical value -3 and the term containing the positive exponent as it is and consider the term containing the negative exponent, i.e., \[{{b}^{-5}}\], use the basic formula of the topic ‘exponents and powers’ given as: - \[{{a}^{-m}}=\dfrac{1}{{{a}^{m}}}\] to simplify the expression such that it contains only positive exponents.

Complete step by step solution:
Here, we have been provided with the expression \[-3{{a}^{2}}{{b}^{-5}}\] and we are asked to simplify this expression so that the exponents become positive. So, here we are going to use some basic properties and formulas of the topic ‘exponents and powers’.
Now, let us assume the given expression as E, so we have,
\[\Rightarrow E=-3{{a}^{2}}{{b}^{-5}}\]
Clearly, we can see that here we have three different components in this expression, they are: - \[-3,{{a}^{2}}\] and \[{{b}^{-5}}\], so we can write it as: -
\[\Rightarrow E=\left( -3 \right)\times \left( {{a}^{2}} \right)\times \left( {{b}^{-5}} \right)\]
Here, we can write (-3) in the exponential form as \[{{\left( -3 \right)}^{1}}\], so we get the expression as: -
\[\Rightarrow E={{\left( -3 \right)}^{1}}\times {{\left( a \right)}^{2}}\times {{\left( b \right)}^{-5}}\]
Clearly, we can see that the exponents of the three components are 1, 2 and -5 respectively where we can conclude that 1 and 2 are positive integers while -5 is negative in nature. So, we need to simplify the component \[{{b}^{-5}}\].
Now, we know that when we have a negative exponent of any number then it is converted into the number having positive exponent by using the formula: - \[{{a}^{-m}}=\dfrac{1}{{{a}^{m}}}\]. Here, m is any positive number. So, using this formula to simplify the expression, we get,
\[\begin{align}
  & \Rightarrow E=\left( -3 \right)\times {{a}^{2}}\times \dfrac{1}{{{b}^{5}}} \\
 & \Rightarrow E=\dfrac{-3{{a}^{2}}}{{{b}^{5}}} \\
\end{align}\]
Since, all the components in the above relation contain positive numbers as their exponent, hence the relation obtained above is our answer.

Note:
You must remember all the basic formulas of the topic ‘exponents and powers’ like: - \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\], \[{{a}^{m}}\div {{a}^{n}}={{a}^{m-n}}\], \[{{\left( {{a}^{m}} \right)}^{n}}={{a}^{m\times n}}\], \[{{a}^{-m}}=\dfrac{1}{{{a}^{m}}}\] etc, as they are used in many other topics. Here, in the above question you cannot make the constant (-3) positive because it is not an exponent. So, do not take its reciprocal and remove the minus sign otherwise it will be considered wrong.