wing exprertons (without using a calculator). rexpecise 26 - the following surds. (1) $$ 25 i^{6} p^{10} $$ 2. $$ \sqrt[3]{8 x^{12} y^{15}} $$ (B) $$ 81 imes \sqrt{322^{3} y^{10}} $$ 4. $$ \sqrt{2 x^{3}} imes \sqrt{8 x^{3}} $$ (5) $$ (8) \sqrt{2} \cdot \sqrt{8} $$ 6. $$ \sqrt[6]{64 a^{6} b^{12} c^{18}} $$ (0) $$ \sqrt[x]{9 x^{10}+16 x^{10}} $$ 8. $$ \sqrt{\frac{27 x^{3} y^{5}}{12 x^{7} y^{3}}} $$ (9) $$ \mathrm{g} \sqrt{-} imes \sqrt{27 m} $$ (10.) $$ \sqrt{2 m^{3}} imes \sqrt{50 m x^{2}} $$ (11) $$ \frac{\sqrt{4 x^{1}}+\sqrt{4 x^{10}}}{5 x^{3}} $$ 12. $$ \sqrt[3]{3 x} imes \sqrt[3]{72 x^{5}} $$

Question

wing exprertons (without using a calculator). rexpecise 26 - the following surds. (1) $$ 25 i^{6} p^{10} $$ 2. $$ \sqrt[3]{8 x^{12} y^{15}} $$ (B) $$ 81 	imes \sqrt{322^{3} y^{10}} $$ 4. $$ \sqrt{2 x^{3}} 	imes \sqrt{8 x^{3}} $$ (5) $$ (8) \sqrt{2} \cdot \sqrt{8} $$ 6. $$ \sqrt[6]{64 a^{6} b^{12} c^{18}} $$ (0) $$ \sqrt[x]{9 x^{10}+16 x^{10}} $$ 8. $$ \sqrt{\frac{27 x^{3} y^{5}}{12 x^{7} y^{3}}} $$ (9) $$ \mathrm{g} \sqrt{-} 	imes \sqrt{27 m} $$ (10.) $$ \sqrt{2 m^{3}} 	imes \sqrt{50 m x^{2}} $$ (11) $$ \frac{\sqrt{4 x^{1}}+\sqrt{4 x^{10}}}{5 x^{3}} $$ 12. $$ \sqrt[3]{3 x} 	imes \sqrt[3]{72 x^{5}} $$

UpStudy AI · Accepted Answer

Let's simplify each of the given expressions step by step.

### 1. $$ 25 i^{6} p^{10} $$

First, we simplify $$ i^{6} $$:
- $$ i^{2} = -1 $$
- $$ i^{4} = 1 $$
- $$ i^{6} = i^{4} \cdot i^{2} = 1 \cdot -1 = -1 $$

Now, substituting back:
$$
25 i^{6} p^{10} = 25 \cdot (-1) \cdot p^{10} = -25 p^{10}
$$

### 2. $$ \sqrt[3]{8 x^{12} y^{15}} $$

We can break this down:
$$
\sqrt[3]{8} = 2, \quad \sqrt[3]{x^{12}} = x^{4}, \quad \sqrt[3]{y^{15}} = y^{5}
$$
Thus,
$$
\sqrt[3]{8 x^{12} y^{15}} = 2 x^{4} y^{5}
$$

### 3. $$ 81 \times \sqrt{32^{3} y^{10}} $$

First, simplify $$ \sqrt{32^{3}} $$:
$$
32 = 2^{5} \implies 32^{3} = (2^{5})^{3} = 2^{15}
$$
So,
$$
\sqrt{32^{3}} = \sqrt{2^{15}} = 2^{7.5} = 2^{7} \cdot \sqrt{2} = 128 \sqrt{2}
$$
Now, substituting back:
$$
81 \times \sqrt{32^{3} y^{10}} = 81 \times 128 \sqrt{2} \times y^{5} = 10368 y^{5} \sqrt{2}
$$

### 4. $$ \sqrt{2 x^{3}} \times \sqrt{8 x^{3}} $$

Using the property of square roots:
$$
\sqrt{2 x^{3}} \times \sqrt{8 x^{3}} = \sqrt{(2)(8)(x^{3})(x^{3})} = \sqrt{16 x^{6}} = 4 x^{3}
$$

### 5. $$ (8) \sqrt{2} \cdot \sqrt{8} $$

First, simplify $$ \sqrt{8} $$:
$$
\sqrt{8} = \sqrt{4 \cdot 2} = 2 \sqrt{2}
$$
Now substituting back:
$$
(8) \sqrt{2} \cdot \sqrt{8} = 8 \sqrt{2} \cdot 2 \sqrt{2} = 16 \cdot 2 = 32
$$

### 6. $$ \sqrt[6]{64 a^{6} b^{12} c^{18}} $$

Breaking it down:
$$
\sqrt[6]{64} = 2, \quad \sqrt[6]{a^{6}} = a, \quad \sqrt[6]{b^{12}} = b^{2}, \quad \sqrt[6]{c^{18}} = c^{3}
$$
Thus,
$$
\sqrt[6]{64 a^{6} b^{12} c^{18}} = 2 a b^{2} c^{3}
$$

### 7. $$ \sqrt{x}{9 x^{10}+16 x^{10}} $$

Combine the terms:
$$
9 x^{10} + 16 x^{10} = 25 x^{10}
$$
Now,
$$
\sqrt{x}{25 x^{10}} = 5 x^{5}
$$

### 8. $$ \sqrt{\frac{27 x^{3} y^{5}}{12 x^{7} y^{3}}} $$

Simplifying the fraction:
$$
\frac{27}{12} = \frac{9}{4}, \quad \frac{x^{3}}{x^{7}} = \frac{1}{x^{4}}, \quad \frac{y^{5}}{y^{3}} = y^{2}
$$
Thus,
$$
\sqrt{\frac{27 x^{3} y^{5}}{12 x^{7} y^{3}}} = \sqrt{\frac{9 y^{2}}{4 x^{4}}} = \frac{3 y}{2 x^{2}}
$$

### 9. $$ g \sqrt{-} \times \sqrt{27 m} $$

Assuming $$ g \sqrt{-1} = g i $$:
$$
g i \cdot \sqrt{27 m} = g i \cdot 3 \sqrt{3} \sqrt{m} = 3 g i \sqrt{3 m}
$$

### 10. $$ \sqrt{2 m^{3}} \times \sqrt{50 m x^{2}} $$

Using the property of square roots:
$$
\sqrt{2 m^{3}} \times \sqrt{50 m x^{2}} = \sqrt{(2)(50)(m^{3})(m)(x^{2})} = \sqrt{100 m^{4} x^{2}} = 10 m^{2} x
$$

### 11. $$ \frac{\sqrt{4 x^{1}}+\sqrt{4 x^{10}}}{5 x^{3}} $$

Simplifying the square roots:
$$
\sqrt{4 x^{1}} = 2 \sqrt{x}, \quad \sqrt{4 x^{10}} = 2 x^{5}
$$
Thus,
$$
\frac{2 \sqrt{x} + 2 x^{5}}{5 x^{3}} = \frac{2(\sqrt{x} + x^{5})}{5 x^{3}}
$$

### 12. $$ \sqrt[3]{3 x} \times \sqrt[3]{72 x^{5}} $$

Using the property of cube roots:
$$
\sqrt[3]{3 x \cdot 72 x^{5}} = \sqrt[3]{216 x^{6}} = \sqrt[3]{216} \cdot \sqrt[3]{x^{6}} = 6 x^{2}
$$

### Summary of Results
1. $$ -25 p^{10} $$
2. $$ 2 x^{4} y^{5} $$
3. $$ 10368 y^{5} \sqrt{2} $$
4. $$ 4 x^{3} $$
5. $$ 32 $$
6. $$ 2 a b^{2} c^{3} $$
7. $$ 5 x^{5} $$
8. $$ \frac{3 y}{2 x^{2}} $$
9. $$ 3 g i \sqrt{3 m} $$
10. $$ 10 m^{2} x $$
11
1. $$ -25 p^{10} $$ 2. $$ 2 x^{4} y^{5} $$ 3. $$ 10368 y^{5} \sqrt{2} $$ 4. $$ 4 x^{3} $$ 5. $$ 32 $$ 6. $$ 2 a b^{2} c^{3} $$ 7. $$ 5 x^{5} $$ 8. $$ \frac{3 y}{2 x^{2}} $$ 9. $$ 3 g i \sqrt{3 m} $$ 10. $$ 10 m^{2} x $$ 11. $$ \frac{2(\sqrt{x} + x^{5})}{5 x^{3}} $$ 12. $$ 6 x^{2} $$

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