Question

Evaluate:
i) ${3^{ - 2}}$
ii) ${\left( { - 3} \right)^{ - 2}}$
iii) ${\left( {\dfrac{1}{2}} \right)^{ - 5}}$

Answer 1

Hint: The given problems are involving indices so we will use the basic rules of indices to solve each problem. We will use powers of basic numbers that are used in the given examples. Take the respective powers and write it in the final position.


Complete step by step solution:
We will use the following identity mainly to solve given problems:
${a^{ - m}} = \dfrac{1}{{{a^m}}}$ … (1)
Where $a$ is a real number and $m$ is an integer.
Now observe that in the given problem all the bases are integers and powers are also integers. So we can use the given identity to solve the problems.
Consider one by one example now.

i) ${3^{ - 2}}$
Here the base is $3$ and index is $ - 2$ .
We will use the identity (1) to proceed further.
${3^{ - 2}} = \dfrac{1}{{{3^2}}}$
We know that ${3^2} = 9$.
Therefore, the given identity is rewritten as:
${3^{ - 2}} = \dfrac{1}{9}$

ii) ${\left( { - 3} \right)^{ - 2}}$
Here the base is $ - 3$ and index is $ - 2$ .
We will use the identity (1) to proceed further.
${\left( { - 3} \right)^{ - 2}} = \dfrac{1}{{{{\left( { - 3} \right)}^2}}}$
We know that ${\left( { - 3} \right)^2} = 9$.
Therefore, the given identity is rewritten as:
${\left( { - 3} \right)^{ - 2}} = \dfrac{1}{9}$

iii) ${\left( {\dfrac{1}{2}} \right)^{ - 5}}$
Here the base is $\dfrac{1}{2}$ and the index is $ - 5$ .
We will use the identity (1) to proceed further.
${\left( {\dfrac{1}{2}} \right)^{ - 5}} = \dfrac{1}{{{{\left( {\dfrac{1}{2}} \right)}^5}}}$
The right-hand side can be rewritten as:
${\left( {\dfrac{1}{2}} \right)^{ - 5}} = {\left( 2 \right)^5}$
We know that ${\left( 2 \right)^5} = 32$.
Therefore, the given identity is rewritten as:
${\left( {\dfrac{1}{2}} \right)^{ - 5}} = 32$


Note: Notice that in the given problems the base need not be necessarily an integer. The identity remains the same whether the base is integer or a real number. The index can also be an integer or a real number. Be careful with the sign while writing the solution.