Question

How do you simplify $2{m^{ - 2}}$ and write it using only positive exponents?

Answer 1

Hint: We know that the above given question is in exponential form. An exponent refers to the number of times a number is multiplied by itself. There is base and exponent or power in this type of equation. Here, in the given question $(2m)$ is the base and the number $ - 2$ is the exponential power. As we know that as per the property of exponent rule if there is $\dfrac{{{a^m}}}{{{a^n}}}$ then it can be written as ${a^{m - n}}$ . When we express a number in exponential form then we can say that it’s power has been raised by the exponent.

Complete step-by-step solution:
There is one basic exponential rule that is commonly used everywhere, ${({a^m})^n} = {a^{m \cdot n}}$.
We should remember these the exponent rule $\dfrac{{{a^m}}}{{{a^n}}} = {a^{m - n}}$ but if the higher power is in the denominator then we can write it as $\dfrac{1}{{{a^{n - m}}}}$. As in the given question we also have an inverse power, there is also another rule we have to apply which is ${a^{ - 1}} = \dfrac{1}{a}$.

To solve exponential equations with base, use the property of power of exponential functions.
Here we have to note that the negative power is only on $m$, not on $2$. So we can write it as $2 \times \dfrac{1}{{{m^2}}}$, since we have to write it using the positive exponent only so we must move the negative power to the denominator to switch its sign.

Hence the required answer of the exponential form is $2 \cdot \dfrac{1}{{{m^2}}}$.

Note: We know that exponential equations are equations in which variables occur as exponents. The formula applied before is true for all real values of $m$ and $n$ . We should solve this kind of problem by using the properties of exponents to simplify the problem. We have to keep in mind that if there is a negative value in the power or exponent then it will reverse the number .i.e. ${m^{ - x}}$ will always be equal to $\dfrac{1}{{{m^x}}}$. We should know that the most commonly used exponential function base is the transcendental number which is denoted by $e$.