If ` ( sqrt(5) )^(7) div( sqrt(5))^(5) = 5^(p)` , then the value of p is

Options

A. 5
B. 2
C. `(3)/(2)`
D. 1

Answer

To solve the equation \( \frac{(\sqrt{5})^7}{(\sqrt{5})^5} = 5^p \), we will follow these steps:

### Step 1: Simplify the left side using the properties of exponents
We know that when dividing powers with the same base, we subtract the exponents. Therefore, we can write:

\[ \frac{(\sqrt{5})^7}{(\sqrt{5})^5} = (\sqrt{5})^{7-5} = (\sqrt{5})^2 \]

### Step 2: Rewrite \((\sqrt{5})^2\) in terms of base 5
Next, we can express \((\sqrt{5})^2\) in terms of base 5. We know that:

\[ \sqrt{5} = 5^{1/2} \]

Thus,

\[ (\sqrt{5})^2 = (5^{1/2})^2 \]

### Step 3: Apply the power of a power property
Using the property of exponents that states \((a^m)^n = a^{m \cdot n}\), we have:

\[ (5^{1/2})^2 = 5^{(1/2) \cdot 2} = 5^1 \]

### Step 4: Set the exponents equal to each other
Now we can set the left side equal to the right side of the original equation:

\[ 5^1 = 5^p \]

### Step 5: Solve for \(p\)
Since the bases are the same, we can equate the exponents:

\[ 1 = p \]

Thus, the value of \(p\) is:

\[ \boxed{1} \]

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