Let's solve both parts of this problem step by step.

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Part (a): Base Two to Base Ten Conversion

We are given a binary number $x = 1.1011 \times 2^{\text{two} 110}$. The exponent $110$ is in binary, so let's break it down.

Step 1: Convert the exponent from binary to decimal.

The exponent $110_2$ in binary equals $6_{10}$ in decimal. This means that the number is $1.1011_2 \times 2^6$.

Step 2: Convert the binary fraction $1.1011_2$ to decimal.

The number $1.1011_2$ can be broken into its components:

$1.1011_2 = 1 \times 2^0 + 1 \times 2^{-1} + 0 \times 2^{-2} + 1 \times 2^{-3} + 1 \times 2^{-4}$

This simplifies to:

$1.1011_2 = 1 + \frac{1}{2} + \frac{0}{4} + \frac{1}{8} + \frac{1}{16} = 1 + 0.5 + 0 + 0.125 + 0.0625 = 1.6875$

So, $1.1011_2 = 1.6875_{10}$ in decimal.

Step 3: Multiply by $2^6$.

Now, multiply $1.6875_{10}$ by $2^6$:

$1.6875 \times 64 = 108$

Thus, the base ten representation of $x$ is $\boxed{108}$.

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Part (b): Representing $\frac{57}{4}$ in Base Four

We need to convert the fraction $\frac{57}{4}$ to base four.

Step 1: Simplify $\frac{57}{4}$.

First, simplify the fraction:

$\frac{57}{4} = 14.25$

Step 2: Convert the integer part (14) to base four.

To convert $14_{10}$ to base four, divide 14 by 4:

$14 \div 4 = 3 \text{ remainder } 2$ $3 \div 4 = 0 \text{ remainder } 3$

So, $14_{10} = 32_4$.

Step 3: Convert the decimal part (0.25) to base four.

Now, convert the decimal part $0.25_{10}$ to base four. Multiply $0.25$ by 4:

$0.25 \times 4 = 1$

Thus, $0.25_{10} = 1_4$.

Step 4: Combine the results.

The base four representation of $14.25_{10}$ is $32.1_4$.

Thus, $\frac{57}{4}$ is represented in base four as $\boxed{32.1_4}$.

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Summary:

• (a) The base ten representation of $1.1011 \times 2^{110_2}$ is $108$.
• (b) The base four representation of $\frac{57}{4}$ is $32.1_4$.

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Would you like more details on any part of the solution or further explanation?

Here are five questions to explore this topic further:

1. How do you convert a binary number with a fractional part to base ten?
2. Why is $110_2$ equal to $6_{10}$ in decimal?
3. What are the steps to convert a decimal fraction to a different base?
4. How does the base-four system differ from the base-ten system in representing fractions?
5. Can you generalize the method for converting binary numbers with floating-point notation to other bases?

Tip: When converting fractions to other bases, multiplying the fractional part repeatedly by the base helps extract the digits after the point!