If m and p are positive integer and `(2sqrt(2))^(m)=32^(n)`, what is the value of `(p)/(m)`?

Answer

To solve the equation \((2\sqrt{2})^m = 32^p\) and find the value of \(\frac{p}{m}\), we will follow these steps:

### Step 1: Rewrite the left-hand side
We start with the left-hand side of the equation:
\[ (2\sqrt{2})^m \]
We can express \(\sqrt{2}\) as \(2^{1/2}\). Therefore, we have:
\[ 2\sqrt{2} = 2 \cdot 2^{1/2} = 2^{1 + 1/2} = 2^{3/2} \]
Thus, we can rewrite the left-hand side:
\[ (2\sqrt{2})^m = (2^{3/2})^m \]

### Step 2: Apply the power of a power property
Using the property of exponents \((a^b)^c = a^{bc}\), we can simplify:
\[ (2^{3/2})^m = 2^{(3/2)m} = 2^{\frac{3m}{2}} \]

### Step 3: Rewrite the right-hand side
Next, we rewrite the right-hand side:
\[ 32^p \]
We know that \(32\) can be expressed as a power of \(2\):
\[ 32 = 2^5 \]
Thus, we can rewrite the right-hand side:
\[ 32^p = (2^5)^p = 2^{5p} \]

### Step 4: Set the exponents equal
Now we have:
\[ 2^{\frac{3m}{2}} = 2^{5p} \]
Since the bases are the same, we can set the exponents equal to each other:
\[ \frac{3m}{2} = 5p \]

### Step 5: Solve for \(\frac{p}{m}\)
To find \(\frac{p}{m}\), we can rearrange the equation:
\[ 5p = \frac{3m}{2} \]
Dividing both sides by \(m\) and \(5\):
\[ p = \frac{3m}{10} \]
Now, dividing both sides by \(m\):
\[ \frac{p}{m} = \frac{3}{10} \]

### Final Answer
Thus, the value of \(\frac{p}{m}\) is:
\[ \frac{p}{m} = \frac{3}{10} \]