If `16timesa^(2)times64=(4timesb)^(2)` and a and b are positive integers, then b is how many times greater than a?

Answer

To solve the equation \( 16 \times a^2 \times 64 = (4 \times b)^2 \) and find out how many times greater \( b \) is than \( a \), we can follow these steps:

### Step 1: Simplify the left side of the equation
First, we simplify the left side of the equation:
\[ 16 \times 64 = 1024 \]
So, we rewrite the equation as:
\[ 1024 \times a^2 = (4 \times b)^2 \]

### Step 2: Expand the right side of the equation
Next, we expand the right side:
\[ (4 \times b)^2 = 16b^2 \]
Now, the equation looks like this:
\[ 1024 \times a^2 = 16b^2 \]

### Step 3: Rearrange the equation
We can rearrange the equation to isolate \( b^2 \):
\[ b^2 = \frac{1024 \times a^2}{16} \]

### Step 4: Simplify the fraction
Now, we simplify the fraction:
\[ b^2 = 64 \times a^2 \]

### Step 5: Take the square root of both sides
Taking the square root of both sides gives us:
\[ b = \sqrt{64 \times a^2} \]
This simplifies to:
\[ b = 8a \]

### Step 6: Determine how many times greater \( b \) is than \( a \)
To find how many times greater \( b \) is than \( a \), we can express this as:
\[ \frac{b}{a} = \frac{8a}{a} = 8 \]

Thus, \( b \) is 8 times greater than \( a \).

### Final Answer:
\( b \) is 8 times greater than \( a \).

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