Question

How do you simplify and write \[{{x}^{-7}}\] with positive exponents?

Answer 1

Hint: We should know the properties of exponents which states that, \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]. And another property \[{{a}^{0}}=1\], here \[a,m\And n\in \] Real numbers. To write the expression of the form \[{{a}^{-m}}\] with positive exponents. The exponent term should be taken to the denominator. This can be done by multiplying and dividing the expression with an exponential term having the same base, but negative exponent as that of the original term. Which means it should be multiplied \[{{a}^{m}}\].

Complete answer:
The given exponential term is \[{{x}^{-7}}\], we have to write this with a positive exponent. The expression is of the form \[{{a}^{-m}}\]. Here a = x and m = 7. To write \[{{a}^{-m}}\] with a positive exponent, we have to multiply and divide it by \[{{a}^{m}}\]. Hence, to write \[{{x}^{-7}}\] with a positive exponent, we have to multiply and divide it by the term \[{{x}^{7}}\]. By doing this we get,
\[\Rightarrow {{x}^{-7}}\times \dfrac{{{x}^{7}}}{{{x}^{7}}}=\dfrac{{{x}^{-7}}\times {{x}^{7}}}{{{x}^{7}}}\]
Using the property of exponents which states that, \[{{a}^{m}}\times {{a}^{n}}={{a}^{m+n}}\]. The above expression can be written as,
\[\Rightarrow \dfrac{{{x}^{-7+7}}}{{{x}^{7}}}=\dfrac{{{x}^{0}}}{{{x}^{7}}}\]
We know the property of exponents which state that, \[{{a}^{0}}=1\] using this property in the above expression we get
\[\Rightarrow \dfrac{{{x}^{0}}}{{{x}^{7}}}=\dfrac{1}{{{x}^{7}}}\]
The above expression has a positive exponent at the denominator.
Hence, the positive exponent form of ${x}^{-7}$ is \[\dfrac{1}{{{x}^{7}}}\].

Note: For these types of problems, the properties of exponents should be remembered. For example, multiplication and division of exponents with the same base, exponents with power zero. We can do a similar thing for writing an exponential term having a positive exponent, in the form of having a negative exponent.