Write the denominator of the rational number ` ( 771)/( 3000)`in the form ` 2 ^(p) 5 ^(q)` , where p and q are non - negative integers

Answer

To express the denominator of the rational number \( \frac{771}{3000} \) in the form \( 2^p \cdot 5^q \), where \( p \) and \( q \) are non-negative integers, we can follow these steps:

### Step 1: Factor the denominator
We start with the denominator, which is 3000. We need to factor 3000 into its prime factors.

\[ 3000 = 3 \times 1000 \]

### Step 2: Factor 1000
Next, we factor 1000 into its prime factors.

\[ 1000 = 10 \times 100 \]
\[ 10 = 2 \times 5 \]
\[ 100 = 10 \times 10 = (2 \times 5) \times (2 \times 5) = 2^2 \times 5^2 \]

Combining these, we have:

\[ 1000 = 2^3 \times 5^3 \]

### Step 3: Combine the factors
Now, substituting back into our factorization of 3000:

\[ 3000 = 3 \times (2^3 \times 5^3) \]

### Step 4: Identify the form \( 2^p \cdot 5^q \)
In our factorization, we can see that the denominator \( 3000 \) has the prime factors \( 2^3 \) and \( 5^3 \). The factor of 3 does not affect the representation in the form \( 2^p \cdot 5^q \).

Thus, we can write:

\[ 3000 = 2^3 \cdot 5^3 \cdot 3 \]

### Step 5: Determine \( p \) and \( q \)
From the expression \( 2^3 \cdot 5^3 \), we can identify:

- \( p = 3 \)
- \( q = 3 \)

### Final Answer
Thus, the denominator \( 3000 \) can be expressed in the form \( 2^p \cdot 5^q \) where \( p = 3 \) and \( q = 3 \).