Without using a calculator, find the value of: $$ \begin{array}{ll} ext { a) } \frac{\sqrt{75}+\sqrt{48}}{\sqrt{12}} & ext { b) } \sqrt{3}(\sqrt{48}-3 \sqrt{75}+2 \sqrt{108} \ ext { a) } \sqrt{50}+\sqrt{32}-\sqrt{18} & ext { d) } \sqrt[3]{\sqrt{64}}\end{array} $$

Question

Without using a calculator, find the value of: $$ \begin{array}{ll}	ext { a) } \frac{\sqrt{75}+\sqrt{48}}{\sqrt{12}} & 	ext { b) } \sqrt{3}(\sqrt{48}-3 \sqrt{75}+2 \sqrt{108} \ 	ext { a) } \sqrt{50}+\sqrt{32}-\sqrt{18} & 	ext { d) } \sqrt[3]{\sqrt{64}}\end{array} $$

UpStudy AI · Accepted Answer

Let's simplify each expression step by step without using a calculator.

---

### **a) $$ \frac{\sqrt{75} + \sqrt{48}}{\sqrt{12}} $$**

**Step 1: Simplify each square root by factoring out perfect squares.**

- $$ \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3} $$
- $$ \sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4\sqrt{3} $$
- $$ \sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3} $$

**Step 2: Substitute the simplified forms back into the expression.**

$$
\frac{\sqrt{75} + \sqrt{48}}{\sqrt{12}} = \frac{5\sqrt{3} + 4\sqrt{3}}{2\sqrt{3}}
$$

**Step 3: Combine like terms in the numerator.**

$$
\frac{9\sqrt{3}}{2\sqrt{3}}
$$

**Step 4: Simplify by canceling $$ \sqrt{3} $$.**

$$
\frac{9}{2}
$$

**Final Answer:** $$ \frac{9}{2} $$ or **4.5**

---

### **b) $$ \sqrt{3} (\sqrt{48} - 3 \sqrt{75} + 2 \sqrt{108}) $$**

**Step 1: Simplify each square root inside the parentheses.**

- $$ \sqrt{48} = 4\sqrt{3} $$ (as shown in part a)
- $$ \sqrt{75} = 5\sqrt{3} $$ (as shown in part a)
- $$ \sqrt{108} = \sqrt{36 \times 3} = 6\sqrt{3} $$

**Step 2: Substitute the simplified forms back into the expression.**

$$
\sqrt{3} \left( 4\sqrt{3} - 3 \times 5\sqrt{3} + 2 \times 6\sqrt{3} \right) 
$$

$$
= \sqrt{3} \left( 4\sqrt{3} - 15\sqrt{3} + 12\sqrt{3} \right)
$$

**Step 3: Combine like terms inside the parentheses.**

$$
= \sqrt{3} \left( (4 - 15 + 12)\sqrt{3} \right)
$$

$$
= \sqrt{3} \times 1\sqrt{3} = \sqrt{3} \times \sqrt{3}
$$

**Step 4: Multiply the square roots.**

$$
\sqrt{3} \times \sqrt{3} = \sqrt{9} = 3
$$

**Final Answer:** **3**

---

### **c) $$ \sqrt{50} + \sqrt{32} - \sqrt{18} $$**

**Step 1: Simplify each square root by factoring out perfect squares.**

- $$ \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} $$
- $$ \sqrt{32} = \sqrt{16 \times 2} = 4\sqrt{2} $$
- $$ \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2} $$

**Step 2: Substitute the simplified forms back into the expression.**

$$
5\sqrt{2} + 4\sqrt{2} - 3\sqrt{2}
$$

**Step 3: Combine like terms.**

$$
(5 + 4 - 3)\sqrt{2} = 6\sqrt{2}
$$

**Final Answer:** $$ 6\sqrt{2} $$

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### **d) $$ \sqrt[3]{\sqrt{64}} $$**

**Step 1: Simplify the inner square root.**

- $$ \sqrt{64} = 8 $$ because $$ 8 \times 8 = 64 $$

**Step 2: Compute the cube root of 8.**

- $$ \sqrt[3]{8} = 2 $$ because $$ 2 \times 2 \times 2 = 8 $$

**Final Answer:** **2**

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a) 4.5 b) 3 c) $$ 6\sqrt{2} $$ d) 2

Without using a calculator, find the value of:

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Solución

Beyond the Answer

Part a:

Part b:

Part c:

Part d:

Final Answers:

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