Suppose a base 4 place-value system has its digits represented as shown to the right. a. Determine the value of RGBRG, in base 10. b. Write 180 in the base 4 system using only the symbols shown to the right. a. The value of RGBRG¸ in base 10 is (Simplify your answer.)

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### Understanding a Base-4 Place-Value System

In this example, we will learn how to interpret and convert numbers from a base-4 place-value system to the more familiar base-10 system.

#### Base-4 System Representation:

The digits in the base-4 system are represented by the following symbols:
- \( \textcolor{blue}{B} \) = 0
- \( \textcolor{red}{R} \) = 1
- \( \textcolor{orange}{O} \) = 2
- \( \textcolor{green}{G} \) = 3

### Exercises

**a. Determine the value of \( \textcolor{red}{R} \textcolor{green}{G} \textcolor{blue}{B} \textcolor{red}{R} \textcolor{green}{G} \) in base 10.**

In this step, convert the given base-4 number \( \textcolor{red}{R} \textcolor{green}{G} \textcolor{blue}{B} \textcolor{red}{R} \textcolor{green}{G} \) to base 10.

**b. Write 180 in the base-4 system using only the symbols shown to the right.**

Convert the base-10 number 180 to the base-4 system using the symbols \( \textcolor{blue}{B} \), \( \textcolor{red}{R} \), \( \textcolor{orange}{O} \), and \( \textcolor{green}{G} \).

#### Solution for Part (a):

The value of \( \textcolor{red}{R} \textcolor{green}{G} \textcolor{blue}{B} \textcolor{red}{R} \textcolor{green}{G}_4 \) in base 10 is _.

\[ \boxed{\text{Simplify your answer.}} \]
Transcribed Image Text:### Understanding a Base-4 Place-Value System In this example, we will learn how to interpret and convert numbers from a base-4 place-value system to the more familiar base-10 system. #### Base-4 System Representation: The digits in the base-4 system are represented by the following symbols: - \( \textcolor{blue}{B} \) = 0 - \( \textcolor{red}{R} \) = 1 - \( \textcolor{orange}{O} \) = 2 - \( \textcolor{green}{G} \) = 3 ### Exercises **a. Determine the value of \( \textcolor{red}{R} \textcolor{green}{G} \textcolor{blue}{B} \textcolor{red}{R} \textcolor{green}{G} \) in base 10.** In this step, convert the given base-4 number \( \textcolor{red}{R} \textcolor{green}{G} \textcolor{blue}{B} \textcolor{red}{R} \textcolor{green}{G} \) to base 10. **b. Write 180 in the base-4 system using only the symbols shown to the right.** Convert the base-10 number 180 to the base-4 system using the symbols \( \textcolor{blue}{B} \), \( \textcolor{red}{R} \), \( \textcolor{orange}{O} \), and \( \textcolor{green}{G} \). #### Solution for Part (a): The value of \( \textcolor{red}{R} \textcolor{green}{G} \textcolor{blue}{B} \textcolor{red}{R} \textcolor{green}{G}_4 \) in base 10 is _. \[ \boxed{\text{Simplify your answer.}} \]
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