If $$8^{1.2} × 64^a × 40^{2.4} × 25^{1.3} = 40^5$$​, what will be the value of a ?

Correct option is B

Given:

​$$8^{1.2} × 64^a × 40^{2.4} × 25^{1.3} = 40^5$$​​

Concept Used:

$$\begin{array}{|c|c|} \hline \textbf{Operation preference wise} & \textbf{Symbol} \\\hline \text{Brackets} &[],{}, () \\ \hline \text{Orders, of} & (power), √ (root) , of \\ \hline \text{Division}& ÷ \\ \hline \text{Multiplication} & × \\ \hline \text{Addition} & + \\ \hline \text{Subtraction} & - \\\hline \end{array}$$​

Formula Used:

​$$x^m \times x^n = x^{m+n}$$​​

Solution:

​$$8^{1.2} × 64^a × 40^{2.4} × 25^{1.3} = 40^5$$​​

​$$(2^3)^{1.2} \times (2^6)^a \times (2^3 \times 5^1)^{2.4} \times (5^2)^{1.3} = (2^3 \times 5^1)^5$$

$$2^{3.6} \times 2^{6a} \times 2^{7.2} \times 5^{2.4} \times 5^{2.6} = 2^{15} \times 5^{5}$$

Grouping powers of 2:​
​$$​​2^{(3.6 + 6a + 7.2)} = 2^{15}$$​​

​3.6 + 6a + 7.2 = 15

6a + 10.8 = 15

6a = 4.2

a = 0.7

Verify for Base 5:

​$$5^{(2.4 + 2.6)} = 5^5$$​​

​$$5^{5} = 5^5$$​ (Which is correct)

Thus, the correct answer is (b).