The decimal representation of `(23)/(2^(3) xx 4^(2))` will be:

Options

A. Terminating
B. Non-terminating
C. Non-terminating and repeating
D. Non-terminating and non-repeating

Answer

To solve the problem of determining the decimal representation of \( \frac{23}{2^3 \times 4^2} \), we will follow these steps:

### Step 1: Simplify the denominator
First, we need to simplify the denominator \( 2^3 \times 4^2 \).

Since \( 4 \) can be expressed as \( 2^2 \), we can rewrite \( 4^2 \) as:
\[ 4^2 = (2^2)^2 = 2^{2 \times 2} = 2^4 \]

Now, substituting this back into the denominator:
\[ 2^3 \times 4^2 = 2^3 \times 2^4 \]

### Step 2: Combine the powers of 2
Using the property of exponents that states \( a^m \times a^n = a^{m+n} \), we can combine the powers:
\[ 2^3 \times 2^4 = 2^{3+4} = 2^7 \]

### Step 3: Rewrite the fraction
Now we can rewrite the original fraction:
\[ \frac{23}{2^3 \times 4^2} = \frac{23}{2^7} \]

### Step 4: Determine if the decimal representation is terminating
A fraction in the form \( \frac{P}{Q} \) has a terminating decimal representation if the prime factorization of \( Q \) contains only the primes 2 and/or 5.

In our case, \( Q = 2^7 \), which consists solely of the prime number 2. Therefore, the decimal representation of \( \frac{23}{2^7} \) will be terminating.

### Step 5: Calculate the decimal value
Next, we can calculate the decimal value of \( \frac{23}{2^7} \):
\[ 2^7 = 128 \]
Thus,
\[ \frac{23}{128} \]

Now, we can perform the division:
\[ 23 \div 128 \approx 0.1796875 \]

### Final Answer
The decimal representation of \( \frac{23}{2^3 \times 4^2} \) is \( 0.1796875 \), which is a terminating decimal.

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