Question

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

Answer 1

Hint: First of all, we will use the fact that $a^{-x}={\displaystyle \frac{1}{a^x}}$. 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: $a^{-x}={\displaystyle \frac{1}{a^x}}$. ………..(2)
Replacing a by 3 and x by 2 in the equation number 2, we will then obtain the following equation with us:-
$\Rightarrow 3^{-2}={\displaystyle \frac{1}{3^2}}$
We can write the above expression as follows:-
$\Rightarrow 3^{-2}={\displaystyle \frac{1}{3\times 3}}$
Simplifying the denominator of the right hand side of the above expression, we will then obtain the following expression with us:-
$\Rightarrow 3^{-2}={\displaystyle \frac{1}{9}}$ ………………(3)
Replacing a by 2 and x by 3 in the equation number 2, we will then obtain the following equation with us:-
$\Rightarrow 2^{-3}={\displaystyle \frac{1}{2^3}}$
We can write the above expression as follows:-
$\Rightarrow 2^{-3}={\displaystyle \frac{1}{2\times 2\times 2}}$
Simplifying the denominator of the right hand side of the above expression, we will then obtain the following expression with us:-
$\Rightarrow 2^{-3}={\displaystyle \frac{1}{8}}$ ………………(4)
Putting the equations number 3 and 4 in the equation number 1, we will then obtain the following expression with us:-
$\Rightarrow 3^{-2}+2^{-3}={\displaystyle \frac{1}{9}}+{\displaystyle \frac{1}{8}}$
Taking the least common multiple of 8 and 9, we have the following expression with us:-
$\Rightarrow 3^{-2}+2^{-3}={\displaystyle \frac{8+9}{72}}={\displaystyle \frac{17}{72}}$
Hence, we have the required answer.

Note: The students must note that we have made use of the fact that $a^{-x}={\displaystyle \frac{1}{a^x}}$ 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.