QuestionWrite $2212_{\text{three}}$ in expanded form and convert it to base ten. Choose the correct expanded form below.

Studdy Solution

STEP 1

Assumptions1. The given number is $221_{\text {three }}$ . We need to write this number in expanded form3. We need to convert this number to base ten

STEP 2

To write the number in expanded form, we need to multiply each digit by the base (which is in this case) raised to the power of its position, counting from right to left and starting with0.
The expanded form of a number in base $b$ is given by $d_n \cdot b^n + d_{n-1} \cdot b^{n-1} + \ldots + d2 \cdot b^2 + d1 \cdot b^1 + d0 \cdot b^0$ where $d_n, d_{n-1}, \ldots, d2, d1, d0$ are the digits of the number from left to right.

STEP 3

Now, plug in the given values for the digits and the base to write the number $2212_{\text {three }}$ in expanded form.
$Expanded\, form =2 \cdot3^3 +2 \cdot3^2 +1 \cdot3^1 +2 \cdot3^0$

STEP 4

Now, we need to convert the number to base ten. We can do this by evaluating the expression we obtained in the previous step.
$Base\, ten\, number =2 \cdot3^3 +2 \cdot3^2 +1 \cdot3^1 +2 \cdot3^0$

STEP 5

Calculate the base ten number.
$Base\, ten\, number =2 \cdot27 +2 \cdot9 +1 \cdot3 +2 \cdot1 =54 +18 +3 +2 =77$ The expanded form of $2212_{\text {three }}$ is $2 \cdot3^3 +2 \cdot3^2 +1 \cdot3^1 +2 \cdot3^0$ and its base ten equivalent is77. So, the correct answer is D.

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