Simplify the following expressions: (i) [5(8^(1/3) + 27^(1/3))]^(1/4) (ii) sqrt(45) - 3sqrt(20) + 4sqrt(5) (iii) sqrt(24)/8 + sqrt(54)/9

Here are the simplifications for each expression:

(i) $\left[5\left(8^{\frac{1}{3}} + 27^{\frac{1}{3}}\right)\right]^{\frac{1}{4}}$

Step 1: Evaluate the fractional exponents.

$$8^{\frac{1}{3}} = \sqrt[3]{8} = 2$$ $$27^{\frac{1}{3}} = \sqrt[3]{27} = 3$$

Step 2: Substitute these values into the expression.

$$\left[5(2 + 3)\right]^{\frac{1}{4}}$$

Step 3: Simplify inside the brackets.

$$\left[5(5)\right]^{\frac{1}{4}} = [25]^{\frac{1}{4}}$$

Step 4: Evaluate the final fractional exponent.

$$25^{\frac{1}{4}} = (5^2)^{\frac{1}{4}} = 5^{2 \times \frac{1}{4}} = 5^{\frac{1}{2}} = \sqrt{5}$$

The simplified expression is $\boxed{\sqrt{5}}$.

(ii) $\sqrt{45} - 3\sqrt{20} + 4\sqrt{5}$

Step 1: Simplify each square root.

$$\sqrt{45} = \sqrt{9 \times 5} = 3\sqrt{5}$$ $$\sqrt{20} = \sqrt{4 \times 5} = 2\sqrt{5}$$

Step 2: Substitute the simplified roots back into the expression.

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

Step 3: Perform the multiplication and combine like terms.

$$3\sqrt{5} - 6\sqrt{5} + 4\sqrt{5} = (3 - 6 + 4)\sqrt{5} = 1\sqrt{5} = \sqrt{5}$$

The simplified expression is $\boxed{\sqrt{5}}$.

(iii) $\frac{\sqrt{24}}{8} + \frac{\sqrt{54}}{9}$

Step 1: Simplify each square root.

$$\sqrt{24} = \sqrt{4 \times 6} = 2\sqrt{6}$$ $$\sqrt{54} = \sqrt{9 \times 6} = 3\sqrt{6}$$

Step 2: Substitute the simplified roots into the expression.

$$\frac{2\sqrt{6}}{8} + \frac{3\sqrt{6}}{9}$$

Step 3: Simplify the fractions.

$$\frac{\sqrt{6}}{4} + \frac{\sqrt{6}}{3}$$

Step 4: Find a common denominator and add the fractions. The common denominator for 4 and 3 is 12.

$$\frac{3\sqrt{6}}{12} + \frac{4\sqrt{6}}{12} = \frac{3\sqrt{6} + 4\sqrt{6}}{12} = \frac{7\sqrt{6}}{12}$$

The simplified expression is $\boxed{\frac{7\sqrt{6}}{12}}$.

(iv) $\sqrt[4]{12} \times \sqrt[6]{7}$

Step 1: Convert the roots to a common index. The least common multiple (LCM) of 4 and 6 is 12.

$$\sqrt[4]{12} = 12^{\frac{1}{4}} = 12^{\frac{3}{12}} = \sqrt[12]{12^3} = \sqrt[12]{1728}$$ $$\sqrt[6]{7} = 7^{\frac{1}{6}} = 7^{\frac{2}{12}} = \sqrt[12]{7^2} = \sqrt[12]{49}$$

Step 2: Multiply the roots with the common index.

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