Question

How do you simplify ${3^{ - 2}} + {2^{ - 3}}$?

Answer 1

Hint: First of all, we will use the fact that ${a^{ - x}} = \dfrac{1}{{{a^x}}}$. Then, we will just take the least common multiple of the numbers in the denominators and thus, we will have the required simplification.

Complete step by step solution:
We are given that we are required to simplify ${3^{ - 2}} + {2^{ - 3}}$. ……………(1)
We know that for any real number a and x, we have: ${a^{ - x}} = \dfrac{1}{{{a^x}}}$. ………..(2)
Replacing a by 3 and x by 2 in the equation number 2, we will then obtain the following equation with us:-
$ \Rightarrow {3^{ - 2}} = \dfrac{1}{{{3^2}}}$
We can write the above expression as follows:-
$ \Rightarrow {3^{ - 2}} = \dfrac{1}{{3 \times 3}}$
Simplifying the denominator of the right hand side of the above expression, we will then obtain the following expression with us:-
$ \Rightarrow {3^{ - 2}} = \dfrac{1}{9}$ ………………(3)
Replacing a by 2 and x by 3 in the equation number 2, we will then obtain the following equation with us:-
$ \Rightarrow {2^{ - 3}} = \dfrac{1}{{{2^3}}}$
We can write the above expression as follows:-
$ \Rightarrow {2^{ - 3}} = \dfrac{1}{{2 \times 2 \times 2}}$
Simplifying the denominator of the right hand side of the above expression, we will then obtain the following expression with us:-
$ \Rightarrow {2^{ - 3}} = \dfrac{1}{8}$ ………………(4)
Putting the equations number 3 and 4 in the equation number 1, we will then obtain the following expression with us:-
$ \Rightarrow {3^{ - 2}} + {2^{ - 3}} = \dfrac{1}{9} + \dfrac{1}{8}$
Taking the least common multiple of 8 and 9, we have the following expression with us:-
$ \Rightarrow {3^{ - 2}} + {2^{ - 3}} = \dfrac{{8 + 9}}{{72}} = \dfrac{{17}}{{72}}$
Hence, we have the required answer.

Note: The students must note that we have made use of the fact that ${a^{ - x}} = \dfrac{1}{{{a^x}}}$ for real numbers a and x.
The students must also note that ${a^n}$ means that we are multiplying a, n times with itself.
So we have the following expression with us:-
$ \Rightarrow {a^n} = a \times a \times a \times .......... \times a$ ( n times )
The students must also note that the least common multiple of 9 and 8 is the multiplication of 9 and 8 because H C F ( 9, 8) = 1.