Question

How do you simplify each expression using positive exponents \[{{m}^{3}}{{n}^{-6}}{{p}^{0}}\]?

Answer 1

Hint: Using the exponent properties we will simplify the given expression. We will deal with the three components individually and then write them together. ${{m}^{3}}$ is in the simplified form, then we have \[{{n}^{-6}}\] which we can write it as \[\dfrac{1}{{{n}^{6}}}\], as per the properties of exponents and then the \[{{p}^{0}}\] which equals 1. When we write the simplified forms of these components together, we have the simplified form of the expression.

Complete step by step solution:
According to the given question, we have an expression which we have to simplify as much as possible. So for the simplification, we will be using the exponential properties.
We have the given expression as,
 \[{{m}^{3}}{{n}^{-6}}{{p}^{0}}\]
We have three components in the given expression and so we will simplify each terms separately and then combine together for the final answer.
Firstly, we have, ${{m}^{3}}$, but we can see it is already in the simplified form and could no more be simplified. We will move on with the next term.
Next we have, \[{{n}^{-6}}\]. As per one of the properties of exponents, we know that, \[{{a}^{-n}}=\dfrac{1}{{{a}^{n}}}\].
So we get, \[{{n}^{-6}}=\dfrac{1}{{{n}^{6}}}\].
Then, we have, \[{{p}^{0}}\]. We know that any number with the power zero becomes nothing but 1. That is, \[{{p}^{0}}=1\].
We will combine the three components that we simplified separately and we get,
\[{{m}^{3}}{{n}^{-6}}{{p}^{0}}\]
\[\Rightarrow {{m}^{3}}\left( \dfrac{1}{{{n}^{6}}} \right)(1)\]
\[\Rightarrow \dfrac{{{m}^{3}}}{{{n}^{6}}}\]

Therefore, the simplified form of the given expression is \[\dfrac{{{m}^{3}}}{{{n}^{6}}}\].

Note: The use of the exponential properties should be done carefully and only after the required conditions are met. If the number or a variable has a power zero, then it is a straight away 1, that is, \[{{a}^{0}}={{5}^{0}}=1\]. Also, the property \[{{a}^{-n}}=\dfrac{1}{{{a}^{n}}}\], could be used in the opposite way as well, that is, \[{{a}^{n}}=\dfrac{1}{{{a}^{-n}}}\] when required.