If ?% of `(4)^7 =512` then what will come in place of interrogative sign (?)

Options

A. 25/8
B. 5
C. 8
D. 10

Answer

To solve the equation \( ?\% \text{ of } (4)^7 = 512 \), we will follow these steps:

### Step 1: Rewrite the equation
We can express the percentage as a fraction:
\[ \frac{x}{100} \cdot (4)^7 = 512 \]
where \( x \) is the value we need to find.

### Step 2: Calculate \( (4)^7 \)
We know that \( 4 = 2^2 \), so:
\[ (4)^7 = (2^2)^7 = 2^{14} \]
Thus, we can rewrite the equation as:
\[ \frac{x}{100} \cdot 2^{14} = 512 \]

### Step 3: Rewrite 512 in terms of powers of 2
Next, we express 512 as a power of 2:
\[ 512 = 2^9 \]
Now, substituting this back into our equation gives us:
\[ \frac{x}{100} \cdot 2^{14} = 2^9 \]

### Step 4: Isolate \( x \)
To isolate \( x \), we can multiply both sides by \( 100 \):
\[ x \cdot 2^{14} = 100 \cdot 2^9 \]
Now, divide both sides by \( 2^{14} \):
\[ x = \frac{100 \cdot 2^9}{2^{14}} \]

### Step 5: Simplify the right side
Using the properties of exponents, we have:
\[ x = 100 \cdot 2^{9 - 14} = 100 \cdot 2^{-5} \]
This simplifies to:
\[ x = \frac{100}{2^5} = \frac{100}{32} = \frac{25}{8} \]

### Conclusion
Thus, the value that comes in place of the interrogative sign \( ? \) is:
\[ \boxed{\frac{25}{8}} \]