Question

How do you simplify ${\left( {\dfrac{3}{4}} \right)^{ - 1}}$ and write it using only positive exponents?

Answer 1

Hint: We have been given a fraction that is raised to the power $ - 1$. We have to convert this negative power into a positive value. For this we will have to use the properties of the exponents. A base raised to negative exponent is equal to the reciprocal of the base raised to the positive exponent.

Complete step-by-step answer:
We have to simplify ${\left( {\dfrac{3}{4}} \right)^{ - 1}}$. We have to write the expression using only positive exponents.
We will use the property of the exponents for negative exponent, which states that
${\left( {\dfrac{a}{b}} \right)^{ - n}} = {\left( {\dfrac{b}{a}} \right)^n}$
i.e. a fraction or number raised to a negative exponent is equal to the reciprocal of the number raised to the absolute value of the exponent.
Thus, using this property of exponents we can write the given expression as,
 \[{\left( {\dfrac{3}{4}} \right)^{ - 1}} = {\left( {\dfrac{4}{3}} \right)^1}\]
Hence, the value of ${\left( {\dfrac{3}{4}} \right)^{ - 1}}$ when written in only positive exponents is \[{\left( {\dfrac{4}{3}} \right)^1}\] .
Alternate method:
To simplify ${\left( {\dfrac{3}{4}} \right)^{ - 1}}$, we can write it as ${\left( {\dfrac{3}{4}} \right)^{0 - 1}}$
We can use a property of exponents which states that ${a^{m - n}} = \dfrac{{{a^m}}}{{{a^n}}}$.
Thus, we can write the given expression as
 ${\left( {\dfrac{3}{4}} \right)^{0 - 1}} = \dfrac{{{{\left( {\dfrac{3}{4}} \right)}^0}}}{{{{\left( {\dfrac{3}{4}} \right)}^1}}}$
We know that any number raised to power $0$ is equal to $1$, i.e. ${a^0} = 1$
Thus we get,
$\dfrac{{{{\left( {\dfrac{3}{4}} \right)}^0}}}{{{{\left( {\dfrac{3}{4}} \right)}^1}}} = \dfrac{1}{{{{\left( {\dfrac{3}{4}} \right)}^1}}} = {\left( {\dfrac{4}{3}} \right)^1}$
Hence, we get the result \[{\left( {\dfrac{3}{4}} \right)^{ - 1}} = {\left( {\dfrac{4}{3}} \right)^1}\]
So, the correct answer is “ \[{\left( {\dfrac{3}{4}} \right)^{ - 1}} = {\left( {\dfrac{4}{3}} \right)^1}\] ”.

Note: A negative exponent only means that the reciprocal of the base is raised to the power that is the absolute value of the negative exponent. It is for the ease of representation. The negative exponent can also be a non-integer number. Also, a negative exponent does not mean that the resulting number will be negative.