EXERCISE 5 Simplify the following without using a calculator: $$ \begin{array}{lll} ext { (a) } \sqrt{7} \sqrt{3} & ext { (b) } \sqrt{7} \sqrt{7}+(\sqrt{11})^{2} & ext { (c) }(2 \sqrt{3})^{2} \ ext { (d) } 3 \sqrt{6}-\sqrt{6}+7 \sqrt{6} & ext { (c) } \sqrt[4]{3}+7 \sqrt[4]{3}-5 \sqrt[4]{3} & ext { (f) } \sqrt{3}+\sqrt{27} \ ext { (g) } 2 \sqrt{18}-\sqrt{32} & ext { (h) } \frac{\sqrt{32}}{\sqrt{2}} & ext { (i) } \frac{\sqrt{51}}{\sqrt{3}}\end{array} $$

Question

EXERCISE 5 Simplify the following without using a calculator: $$ \begin{array}{lll}	ext { (a) } \sqrt{7} \sqrt{3} & 	ext { (b) } \sqrt{7} \sqrt{7}+(\sqrt{11})^{2} & 	ext { (c) }(2 \sqrt{3})^{2} \ 	ext { (d) } 3 \sqrt{6}-\sqrt{6}+7 \sqrt{6} & 	ext { (c) } \sqrt[4]{3}+7 \sqrt[4]{3}-5 \sqrt[4]{3} & 	ext { (f) } \sqrt{3}+\sqrt{27} \ 	ext { (g) } 2 \sqrt{18}-\sqrt{32} & 	ext { (h) } \frac{\sqrt{32}}{\sqrt{2}} & 	ext { (i) } \frac{\sqrt{51}}{\sqrt{3}}\end{array} $$

UpStudy AI · Accepted Answer

Let's simplify each part step-by-step:

### (a) $$ \sqrt{7} \cdot \sqrt{3} $$

Using the property $$ \sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b} $$:

$$
\sqrt{7} \cdot \sqrt{3} = \sqrt{21}
$$

**Answer:** $$ \sqrt{21} $$

---

### (b) $$ \sqrt{7} \cdot \sqrt{7} + (\sqrt{11})^{2} $$

First, simplify each term:

1. $$ \sqrt{7} \cdot \sqrt{7} = \sqrt{7^2} = 7 $$
2. $$ (\sqrt{11})^{2} = 11 $$

Adding them together:

$$
7 + 11 = 18
$$

**Answer:** $$ 18 $$

---

### (c) $$ (2 \sqrt{3})^{2} $$

Square the expression:

$$
(2 \sqrt{3})^{2} = 2^2 \cdot (\sqrt{3})^2 = 4 \cdot 3 = 12
$$

**Answer:** $$ 12 $$

---

### (d) $$ 3 \sqrt{6} - \sqrt{6} + 7 \sqrt{6} $$

Combine like terms:

$$
3 \sqrt{6} - \sqrt{6} + 7 \sqrt{6} = (3 - 1 + 7) \sqrt{6} = 9 \sqrt{6}
$$

**Answer:** $$ 9 \sqrt{6} $$

---

### (e) $$ \sqrt[4]{3} + 7 \sqrt[4]{3} - 5 \sqrt[4]{3} $$

Combine like terms:

$$
\sqrt[4]{3} + 7 \sqrt[4]{3} - 5 \sqrt[4]{3} = (1 + 7 - 5) \sqrt[4]{3} = 3 \sqrt[4]{3}
$$

**Answer:** $$ 3 \sqrt[4]{3} $$

---

### (f) $$ \sqrt{3} + \sqrt{27} $$

Simplify $$ \sqrt{27} $$:

$$
\sqrt{27} = \sqrt{9 \cdot 3} = \sqrt{9} \cdot \sqrt{3} = 3 \sqrt{3}
$$

Add the terms:

$$
\sqrt{3} + 3 \sqrt{3} = 4 \sqrt{3}
$$

**Answer:** $$ 4 \sqrt{3} $$

---

### (g) $$ 2 \sqrt{18} - \sqrt{32} $$

Simplify each radical:

1. $$ \sqrt{18} = \sqrt{9 \cdot 2} = 3 \sqrt{2} $$
2. $$ \sqrt{32} = \sqrt{16 \cdot 2} = 4 \sqrt{2} $$

Now substitute back:

$$
2 \sqrt{18} - \sqrt{32} = 2 \cdot 3 \sqrt{2} - 4 \sqrt{2} = 6 \sqrt{2} - 4 \sqrt{2} = 2 \sqrt{2}
$$

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

---

### (h) $$ \frac{\sqrt{32}}{\sqrt{2}} $$

Simplify the fraction under the square root:

$$
\frac{\sqrt{32}}{\sqrt{2}} = \sqrt{\frac{32}{2}} = \sqrt{16} = 4
$$

**Answer:** $$ 4 $$

---

### (i) $$ \frac{\sqrt{51}}{\sqrt{3}} $$

Simplify the fraction under the square root:

$$
\frac{\sqrt{51}}{\sqrt{3}} = \sqrt{\frac{51}{3}} = \sqrt{17}
$$

**Answer:** $$ \sqrt{17} $$

---

### Summary of Answers

- **(a)** $$ \sqrt{21} $$
- **(b)** $$ 18 $$
- **(c)** $$ 12 $$
- **(d)** $$ 9 \sqrt{6} $$
- **(e)** $$ 3 \sqrt[4]{3} $$
- **(f)** $$ 4 \sqrt{3} $$
- **(g)** $$ 2 \sqrt{2} $$
- **(h)** $$ 4 $$
- **(i)** $$ \sqrt{17} $$
Here are the simplified answers: - **(a)** $$ \sqrt{21} $$ - **(b)** $$ 18 $$ - **(c)** $$ 12 $$ - **(d)** $$ 9 \sqrt{6} $$ - **(e)** $$ 3 \sqrt[4]{3} $$ - **(f)** $$ 4 \sqrt{3} $$ - **(g)** $$ 2 \sqrt{2} $$ - **(h)** $$ 4 $$ - **(i)** $$ \sqrt{17} $$

EXERCISE 5
Simplify the following without using a calculator:

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