What should come in place of the question mark (?) in the following questions:
`(0.04)^5 xx (0.2)^4 ÷ (0.008)^2 = (0.2)^?`

Options

A. 5
B. 6
C. 7
D. 8

Answer

To solve the equation \( (0.04)^5 \times (0.2)^4 \div (0.008)^2 = (0.2)^? \), we will simplify the left-hand side step by step.

### Step 1: Rewrite the numbers in fractional form
First, we can express \( 0.04 \), \( 0.2 \), and \( 0.008 \) in fractional form:
- \( 0.04 = \frac{4}{100} = \frac{1}{25} = \frac{1}{5^2} \)
- \( 0.2 = \frac{2}{10} = \frac{1}{5} \)
- \( 0.008 = \frac{8}{1000} = \frac{1}{125} = \frac{1}{5^3} \)

### Step 2: Substitute the fractions into the equation
Now we can rewrite the equation:
\[ (0.04)^5 = \left(\frac{1}{5^2}\right)^5 = \frac{1^5}{(5^2)^5} = \frac{1}{5^{10}} \]
\[ (0.2)^4 = \left(\frac{1}{5}\right)^4 = \frac{1^4}{5^4} = \frac{1}{5^4} \]
\[ (0.008)^2 = \left(\frac{1}{5^3}\right)^2 = \frac{1^2}{(5^3)^2} = \frac{1}{5^6} \]

### Step 3: Substitute these into the equation
Now substituting these back into the equation gives us:
\[ \frac{1}{5^{10}} \times \frac{1}{5^4} \div \frac{1}{5^6} \]

### Step 4: Simplify the left-hand side
When we divide by a fraction, we multiply by its reciprocal:
\[ \frac{1}{5^{10}} \times \frac{1}{5^4} \times 5^6 = \frac{1 \times 1 \times 5^6}{5^{10} \times 5^4} = \frac{5^6}{5^{10 + 4}} = \frac{5^6}{5^{14}} = 5^{6 - 14} = 5^{-8} \]

### Step 5: Rewrite in terms of \( 0.2 \)
Since \( 0.2 = \frac{1}{5} \), we can express \( 5^{-8} \) as:
\[ 5^{-8} = \left(\frac{1}{5}\right)^8 = (0.2)^8 \]

### Step 6: Conclusion
Thus, we find that:
\[ (0.04)^5 \times (0.2)^4 \div (0.008)^2 = (0.2)^8 \]
This means that the value of the question mark (?) is \( 8 \).

### Final Answer
The answer is \( 8 \).

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