​Select the number that can replace the question mark (?) in the following equation.

​$$47^{7.5} \div 47^{\frac{3}{2}} \times 47^{-3} = (\sqrt{47})^?$$​​

Correct option is A

Given:

​$$47^{7.5} \div 47^{\frac{3}{2}} \times 47^{-3} = (\sqrt{47})^?$$ 

Concept Used:

​$$a^m÷a^n=a^{m-n}$$ 

$$a^m×a^n = a^{m+n}$$ 

$$(a^m)^n = a^{m×n}$$​​

Solution:

L.H.S. 

​$$47^{7.5} \div 47^{\frac{3}{2}} \times 47^{-3} = (\sqrt{47})^?$$ 

$$47^{7.5} \div 47^{\frac{3}{2}} \times 47^{-3} = 47^{7.5 - \frac{3}{2} - 3} = 47^{7.5 - 1.5 - 3} = 47^{3}$$ 

R.H.S.

​$$(\sqrt{47})^? = 47^{\frac{?}{2}}$$

Now, equate the exponents:

​$$47^{3} = 47^{\frac{?}{2}}$$ 

$$3 = \frac{?}{2}$$ 

$$? = 6$$ 

Thus, the answer is 6.