Carry out the addition shown using base-six notation and show that your answer is correct in decimal notation. $$ \begin{array}{ll} ext { (a) } 23_{ ext {six }}+34_{ ext {six }} & ext { (b) } 325_{ ext {six }}+41_{ ext {six }}\end{array} $$ $$ \begin{array}{ll} ext { (a) } 23_{ ext {six }}+34_{ ext {six }}=\square_{ ext {six }} & \end{array} $$

Question

Carry out the addition shown using base-six notation and show that your answer is correct in decimal notation. $$ \begin{array}{ll}	ext { (a) } 23_{	ext {six }}+34_{	ext {six }} & 	ext { (b) } 325_{	ext {six }}+41_{	ext {six }}\end{array} $$ $$ \begin{array}{ll}	ext { (a) } 23_{	ext {six }}+34_{	ext {six }}=\square_{	ext {six }} & \end{array} $$

UpStudy AI · Accepted Answer

We start by performing the addition in base six and then verify the result by converting to decimal.

### (a) $$23_{\text{six}}+34_{\text{six}}$$

1. **Convert each number to decimal:**

- $$23_{\text{six}} = 2 \times 6 + 3 = 12 + 3 = 15_{\text{ten}}$$
   - $$34_{\text{six}} = 3 \times 6 + 4 = 18 + 4 = 22_{\text{ten}}$$

2. **Add in decimal:**

$$
   15_{\text{ten}} + 22_{\text{ten}} = 37_{\text{ten}}
   $$

3. **Convert $$37_{\text{ten}}$$ back to base six:**

- Divide $$37$$ by $$6$$:  
     $$37 \div 6 = 6$$ with a remainder of $$1$$.  
     (This remainder is the units digit in base six.)
   - Divide $$6$$ by $$6$$:  
     $$6 \div 6 = 1$$ with a remainder of $$0$$.  
     (This remainder is the sixes digit.)
   - Divide $$1$$ by $$6$$:  
     $$1 \div 6 = 0$$ with a remainder of $$1$$.  
     (This remainder is the next digit.)

Reading the remainders from last to first, we have:
   
   $$
   37_{\text{ten}} = 101_{\text{six}}
   $$

4. **Verification in decimal:**

$$
   101_{\text{six}} = 1 \times 6^2 + 0 \times 6^1 + 1 \times 6^0 = 36 + 0 + 1 = 37_{\text{ten}}
   $$

This confirms the result.

Thus,

$$
23_{\text{six}}+34_{\text{six}} = 101_{\text{six}}
$$

---

### (b) $$325_{\text{six}}+41_{\text{six}}$$

1. **Convert each number to decimal:**

- $$325_{\text{six}} = 3 \times 6^2 + 2 \times 6 + 5 = 3 \times 36 + 12 + 5 = 108 + 12 + 5 = 125_{\text{ten}}$$
   - $$41_{\text{six}} = 4 \times 6 + 1 = 24 + 1 = 25_{\text{ten}}$$

2. **Add in decimal:**

$$
   125_{\text{ten}} + 25_{\text{ten}} = 150_{\text{ten}}
   $$

3. **Convert $$150_{\text{ten}}$$ to base six:**

- Divide $$150$$ by $$6$$:  
     $$150 \div 6 = 25$$ with a remainder of $$0$$.  
     (This remainder is the units digit.)
   - Divide $$25$$ by $$6$$:  
     $$25 \div 6 = 4$$ with a remainder of $$1$$.  
     (This remainder is the sixes digit.)
   - Divide $$4$$ by $$6$$:  
     $$4 \div 6 = 0$$ with a remainder of $$4$$.  
     (This remainder is the next digit.)

Reading the remainders from last to first gives:
   
   $$
   150_{\text{ten}} = 410_{\text{six}}
   $$

4. **Verification in decimal:**

$$
   410_{\text{six}} = 4 \times 6^2 + 1 \times 6^1 + 0 \times 6^0 = 4 \times 36 + 6 + 0 = 144 + 6 = 150_{\text{ten}}
   $$

This confirms the result.

Thus,

$$
325_{\text{six}}+41_{\text{six}} = 410_{\text{six}}
$$
(a) $$23_{\text{six}} + 34_{\text{six}} = 101_{\text{six}}$$ (b) $$325_{\text{six}} + 41_{\text{six}} = 410_{\text{six}}$$

Carry out the addition shown using base-six notation and show that your answer is correct in decimal notation.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Beyond the Answer

preguntas relacionadas

Latest Arithmetic Questions