Let's go through the math problems you provided:

8. $4^3\sqrt[3]{ab} + 5^3\sqrt[3]{ab}$

Simplify both terms:

• $4^3 = 64$
• $5^3 = 125$

Thus, the expression becomes: [ 64\sqrt[3]{ab} + 125\sqrt[3]{ab} = (64 + 125)\sqrt[3]{ab} = 189\sqrt[3]{ab} ]

9. $\sqrt[3]{40} + \sqrt[3]{135}$

Express each number under the cube roots in terms of prime factors:

• $40 = 8 \times 5 = 2^3 \times 5$
• $135 = 27 \times 5 = 3^3 \times 5$

We can simplify the cube roots: [ \sqrt[3]{40} = \sqrt[3]{2^3 \times 5} = 2\sqrt[3]{5} ] [ \sqrt[3]{135} = \sqrt[3]{3^3 \times 5} = 3\sqrt[3]{5} ]

Thus, the expression becomes: [ 2\sqrt[3]{5} + 3\sqrt[3]{5} = (2 + 3)\sqrt[3]{5} = 5\sqrt[3]{5} ]

10. $5^4\sqrt{80} + \sqrt[3]{16} + 3^4\sqrt[4]{32}$

Simplify each part:

• $5^4 = 625$, and $\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}$ So, $5^4\sqrt{80} = 625 \times 4\sqrt{5} = 2500\sqrt{5}$
• $\sqrt[3]{16} = \sqrt[3]{2^4} = 2\sqrt[3]{2}$
• $3^4 = 81$, and $\sqrt[4]{32} = \sqrt[4]{2^5} = 2\sqrt[4]{2}$ So, $3^4\sqrt[4]{32} = 81 \times 2\sqrt[4]{2} = 162\sqrt[4]{2}$

Thus, the full expression becomes: [ 2500\sqrt{5} + 2\sqrt[3]{2} + 162\sqrt[4]{2} ]

11. $a^2\sqrt{18b^3} + b\sqrt{8a^4b}$

Simplify each part:

• $\sqrt{18b^3} = \sqrt{9 \times 2b^3} = 3b\sqrt{2b}$ So, $a^2\sqrt{18b^3} = a^2 \times 3b\sqrt{2b} = 3a^2b\sqrt{2b}$
• $\sqrt{8a^4b} = \sqrt{4 \times 2a^4b} = 2a^2\sqrt{2b}$ So, $b\sqrt{8a^4b} = b \times 2a^2\sqrt{2b} = 2a^2b\sqrt{2b}$

Thus, the expression becomes: $3a^2b\sqrt{2b} + 2a^2b\sqrt{2b} = 5a^2b\sqrt{2b}$

12. $2\sqrt{20ab^4} + b\sqrt{45ab^2}$

Simplify each part:

• $\sqrt{20ab^4} = \sqrt{4 \times 5ab^4} = 2b^2\sqrt{5a}$ So, $2\sqrt{20ab^4} = 2 \times 2b^2\sqrt{5a} = 4b^2\sqrt{5a}$
• $\sqrt{45ab^2} = \sqrt{9 \times 5ab^2} = 3b\sqrt{5a}$ So, $b\sqrt{45ab^2} = b \times 3b\sqrt{5a} = 3b^2\sqrt{5a}$

Thus, the expression becomes: $4b^2\sqrt{5a} + 3b^2\sqrt{5a} = 7b^2\sqrt{5a}$

Would you like more details on any of these solutions?

Here are some related questions to explore:

1. How can cube roots be simplified using prime factorization?
2. What are the properties of square and cube roots when multiplying radicals?
3. How do you combine like radical terms in an expression?
4. How does the power of a term outside a radical affect the simplification?
5. Why do certain terms in an expression remain under the square root after simplification?

Tip: Always check if a number can be factored into perfect squares or cubes to simplify roots easily.