Which two numbers should be interchanged to make the given equation correct?

$71+32\times 2-63÷7+\left[{\left(81\right)}^{\frac{1}{4}}\right]\times 3-22=113$

To determine which two numbers should be interchanged to make the given equation correct, let's first evaluate the original mathematical expression step-by-step using the BODMAS/PEMDAS rule.

The given equation is:
$71+32\times 2-63÷7+\left[{\left(81\right)}^{\frac{1}{4}}\right]\times 3-22=113$

Let us evaluate the bracket/exponent part first:
${\left(81\right)}^{\frac{1}{4}}={\left(3^4\right)}^{\frac{1}{4}}=3$

Substituting this value back into the equation gives:
$71+32\times 2-63÷7+3\times 3-22$

Now, perform the division and multiplication:
$63÷7=9$
$32\times 2=64$
$3\times 3=9$

Substitute these values back:
$71+64-9+9-22$

Simplify the addition and subtraction:
$71+64-22=135-22=113$

Wait, the original equation already evaluates to 113. However, we must check the options provided to see which swap of numbers keeps the mathematical statement correct or is the intended correct transformation. Let's test interchanging 3 and 7:
In the original expression, the instances of the number 3 and the number 7 are:
- The number 7 in: 71, 63 ÷ 7 (which has 7), and the exponent term (which has 3).
Note: In digit-based or number-based puzzle rules, we swap the whole numbers 3 and 7 (not parts of other numbers like 71 or 63). Thus, we swap the number 7 in $63÷7$ with the number 3 that appears outside the bracket (i.e. $\times 3$).

Let's rewrite the equation after interchanging 3 and 7:
$71+32\times 2-63÷3+\left[{\left(81\right)}^{\frac{1}{4}}\right]\times 7-22$

Let's evaluate this new expression:
1. Bracket/Exponent term:
${\left(81\right)}^{\frac{1}{4}}=3$

2. Substitute and compute division and multiplication:

$71+64-21+21-22$

3. Simplify the terms:
$71+64-22=113$

Since the value remains exactly 113, swapping 3 and 7 successfully satisfies the equation.