8. Multiply the numbers in the indicated bases: $$ 3214_{5} imes 24_{5} $$

Question

8. Multiply the numbers in the indicated bases: $$ 3214_{5} 	imes 24_{5} $$

UpStudy AI · Accepted Answer

Let's solve the problem step by step.

---

### Problem Statement

Multiply the numbers in base 5:
$$
3214_5 \times 24_5
$$

---

### Step 1. Convert Each Number from Base 5 to Decimal

#### For $$3214_5$$:
We use the fact that each digit is weighted by a power of 5:
$$
3214_5 = 3 \times 5^3 + 2 \times 5^2 + 1 \times 5^1 + 4 \times 5^0
$$
Calculate each term:
- $$3 \times 5^3 = 3 \times 125 = 375$$
- $$2 \times 5^2 = 2 \times 25 = 50$$
- $$1 \times 5^1 = 1 \times 5 = 5$$
- $$4 \times 5^0 = 4 \times 1 = 4$$

Thus,
$$
3214_5 = 375 + 50 + 5 + 4 = 434_{10}
$$

#### For $$24_5$$:
Similarly,
$$
24_5 = 2 \times 5^1 + 4 \times 5^0
$$
Calculate each term:
- $$2 \times 5^1 = 2 \times 5 = 10$$
- $$4 \times 5^0 = 4 \times 1 = 4$$

Thus,
$$
24_5 = 10 + 4 = 14_{10}
$$

---

### Step 2. Multiply the Decimal Equivalents

Now multiply the converted numbers:
$$
434 \times 14
$$
Break the multiplication:
$$
434 \times 14 = 434 \times 10 + 434 \times 4 = 4340 + 1736 = 6076_{10}
$$

---

### Step 3. Convert the Decimal Product Back to Base 5

We need to convert $$6076_{10}$$ into base 5 by finding the coefficients for powers of 5.

#### Identify the Highest Power of 5 Needed

Find the greatest power of 5 less than or equal to 6076:
- $$5^5 = 3125$$
- $$5^6 = 15625$$ (too high)

Thus, the highest power we use is $$5^5$$.

#### Determine the Coefficients

1. **Coefficient for $$5^5 = 3125$$:**
   $$
   \text{Coefficient} = \left\lfloor \frac{6076}{3125} \right\rfloor = 1
   $$
   Remainder:
   $$
   6076 - 1 \times 3125 = 2951
   $$

2. **Coefficient for $$5^4 = 625$$:**
   $$
   \text{Coefficient} = \left\lfloor \frac{2951}{625} \right\rfloor = 4
   $$
   Remainder:
   $$
   2951 - 4 \times 625 = 2951 - 2500 = 451
   $$

3. **Coefficient for $$5^3 = 125$$:**
   $$
   \text{Coefficient} = \left\lfloor \frac{451}{125} \right\rfloor = 3
   $$
   Remainder:
   $$
   451 - 3 \times 125 = 451 - 375 = 76
   $$

4. **Coefficient for $$5^2 = 25$$:**
   $$
   \text{Coefficient} = \left\lfloor \frac{76}{25} \right\rfloor = 3
   $$
   Remainder:
   $$
   76 - 3 \times 25 = 76 - 75 = 1
   $$

5. **Coefficient for $$5^1 = 5$$:**
   $$
   \text{Coefficient} = \left\lfloor \frac{1}{5} \right\rfloor = 0
   $$
   Remainder remains:
   $$
   1 - 0 \times 5 = 1
   $$

6. **Coefficient for $$5^0 = 1$$:**
   $$
   \text{Coefficient} = \left\lfloor \frac{1}{1} \right\rfloor = 1
   $$
   Remainder:
   $$
   1 - 1 = 0
   $$

---

### Step 4. Write the Final Answer in Base 5

Collect the coefficients from highest power to lowest:
$$
\text{Coefficient for } 5^5: \, 1 \
\text{Coefficient for } 5^4: \, 4 \
\text{Coefficient for } 5^3: \, 3 \
\text{Coefficient for } 5^2: \, 3 \
\text{Coefficient for } 5^1: \, 0 \
\text{Coefficient for } 5^0: \, 1
$$

So, the product in base 5 is:
$$
143301_5
$$

---

### Final Answer

$$
3214_5 \times 24_5 = 143301_5
$$
$$3214_5 \times 24_5 = 143301_5$$

8. Multiply the numbers in the indicated bases: \( 3214_{5} \times 24_{5} \)

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