9. Simplify: $$ \begin{array}{ll} ext { a. } \sqrt{56} & ext { b. } \sqrt{80}\end{array} $$

Question

9. Simplify: $$ \begin{array}{ll}	ext { a. } \sqrt{56} & 	ext { b. } \sqrt{80}\end{array} $$

UpStudy AI · Accepted Answer

Sure, let's simplify each square root step by step.

---

### a. Simplify $$ \sqrt{56} $$

1. **Factorize 56 into its prime factors:**
   $$
   56 = 8 \times 7 = 4 \times 2 \times 7 = 2^3 \times 7
   $$
   
2. **Identify the largest perfect square factor:**
   - $$ 4 $$ is a perfect square (since $$ 2^2 = 4 $$).
   
3. **Rewrite the square root using the perfect square factor:**
   $$
   \sqrt{56} = \sqrt{4 \times 14} = \sqrt{4} \times \sqrt{14}
   $$
   
4. **Simplify the square root of the perfect square:**
   $$
   \sqrt{4} = 2
   $$
   
5. **Combine the simplified terms:**
   $$
   \sqrt{56} = 2\sqrt{14}
   $$

---

### b. Simplify $$ \sqrt{80} $$

1. **Factorize 80 into its prime factors:**
   $$
   80 = 16 \times 5 = 16 \times 5 = 2^4 \times 5
   $$
   
2. **Identify the largest perfect square factor:**
   - $$ 16 $$ is a perfect square (since $$ 4^2 = 16 $$).
   
3. **Rewrite the square root using the perfect square factor:**
   $$
   \sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5}
   $$
   
4. **Simplify the square root of the perfect square:**
   $$
   \sqrt{16} = 4
   $$
   
5. **Combine the simplified terms:**
   $$
   \sqrt{80} = 4\sqrt{5}
   $$

---

### **Final Answers:**
- **a.** $$ \sqrt{56} = 2\sqrt{14} $$
- **b.** $$ \sqrt{80} = 4\sqrt{5} $$
$$ \sqrt{56} = 2\sqrt{14} $$ and $$ \sqrt{80} = 4\sqrt{5} $$.

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