Question

How do you simplify ${{x}^{-4}}$ and write it using only positive exponents?

Answer 1

Hint: We have been given an exponential function in which a variable-x has a constant negative power equal to -4. We have to change the power of the variable-x from negative to positive. In order to do this, we shall use certain exponential properties. Thus, we shall change the given expression into a fractional form to reverse the sign of the power of the exponent.

Complete step by step solution:
Let us suppose a constant, ‘a’ has to be multiplied by itself ‘b’ times. Then we can write it in the exponential form as ${{a}^{b}}$ instead of writing $a\times a\times a\times a\times ......$upto ‘b’ times.
Exponents have their own set of rules and properties according to which they can be manipulated. One of them is that the sign of the power of the exponent changes if it is taken from the numerator to the denominator or vice versa in a term.
That is, ${{x}^{a}}=\dfrac{1}{{{x}^{-a}}}$.
We have been given the exponent as a function of variable-x, ${{x}^{-4}}$.
In order to reverse the sign of its power, we divide this term by 1 and bring it in the denominator.
\[\Rightarrow {{x}^{-4}}=\dfrac{1}{{{x}^{-\left( -4 \right)}}}\]
\[\Rightarrow {{x}^{-4}}=\dfrac{1}{{{x}^{4}}}\]

Therefore, ${{x}^{-4}}$ can be simplified and written using only positive exponents as $\dfrac{1}{{{x}^{4}}}$.

Note: One of the reasons we use multiplication is because it acts as a shorthand for successive addition. Similarly, exponents act as a shorthand for successive multiplication. If we have to multiply a term (constant or variable) with itself for multiple numbers of times, then we can represent it easily and in a more readable format in the exponential form.