Question

How do you simplify ${9^{ - \dfrac{1}{2}}}$ leaving only positive exponent?

Answer 1

Hint: The power of half always means to take the square root of the number. Negative powers can be solved by interchanging certain aspects of base. Knowledge of fraction and their procedure to reciprocate them will be useful in this question. Reciprocation can be done by changing the sign of the exponent.

Complete step-by-step answer:
We are given the number $9$ to the negative power of $1/2$ . As we know the power of half that is $1/2$ means we have to take the square root of the base to which this power is raised. But we are hindered with a negative sign which is not a big huddle to remove. Any fraction let us say $\dfrac{a}{b}$ can be written opposite which means by exchanging numerator and denominator’s position. In this example ‘a’ will be replaced by ‘b’ and ‘b’ will be replaced by ‘a’. This change can only be done if we put a negative one exponent on the whole fraction after exchanging the numerator and denominator. So our assumed fraction will look like ${\left( {\dfrac{b}{a}} \right)^{ - 1}}$ .
So from this example, the reverse is also true. If we are given a base with a negative exponent, we can make the exponent positive by reversing the numerator and denominator of the base. Also if the base is not a fraction, then it can be reversed by putting $1$ on the numerator and the base on the denominator by making the exponent positive.
Now let us apply this on the expression we have which is ${9^{ - \dfrac{1}{2}}}$
${9^{ - \dfrac{1}{2}}} = \dfrac{1}{{{9^{\dfrac{1}{2}}}}}$
We know power of half means square root so,
$ \Rightarrow {9^{ - \dfrac{1}{2}}} = \dfrac{1}{{\sqrt 9 }}$
On evaluating,
$ \Rightarrow {9^{ - \dfrac{1}{2}}} = \dfrac{1}{3}$
So $\dfrac{1}{3}$ is our required answer.

Note: Be careful of the sign of the exponent while reciprocating the fraction. Since a positive exponent is thought to be of repeated multiplication by the base. In the same way a negative exponent can be thought of repeated division by the base. It is to be noted that polynomials cannot have negative exponents.