$$ \begin{array}{r}30_{ ext {five }}^{\alpha} \ -13_{ ext {five }} \ \hline\end{array} $$

Question

$$ \begin{array}{r}30_{	ext {five }}^{\alpha} \ -13_{	ext {five }} \ \hline\end{array} $$

UpStudy AI · Accepted Answer

We start with the subtraction written in base five:

$$
\begin{array}{r}
30_{\text{five}} \
-13_{\text{five}} \
\hline
\end{array}
$$

Step 1. Convert to decimal (if desired for checking):

- $$30_{\text{five}} = 3 \times 5^1 + 0 = 15$$
- $$13_{\text{five}} = 1 \times 5^1 + 3 = 8$$

Their decimal difference is $$15-8=7$$, and in base five, $$7$$ is written as $$12_{\text{five}}$$ since $$1 \times 5 + 2 = 7$$.

Step 2. Perform the subtraction in base five.

Write the numbers vertically:
$$
\begin{array}{r}
\phantom{1}3\;0 \
-\,\,\,1\;3 \
\hline
\end{array}
$$

Subtract the ones (rightmost) column:
- We need to subtract $$3$$ from $$0$$. Since $$0<3$$ we borrow $$1$$ from the left column. In base five, borrowing $$1$$ adds $$5$$ to the ones column.
- After borrowing, the ones column becomes $$0 + 5 = 5$$ and the tens digit reduces: $$3-1 = 2$$.

Now subtract the ones column:
$$
5-3=2.
$$

Step 3. Now subtract the tens column:
$$
2-1=1.
$$

Thus the result of the subtraction is:
$$
12_{\text{five}}
$$

So, $$\alpha = 12_{\text{five}}$$.
$$ \begin{array}{r} 30_{\text{five}} \ -13_{\text{five}} \ \hline 12_{\text{five}} \end{array} $$ So, $$\alpha = 12_{\text{five}}$$.

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