Let's go through each question one by one.

Problem 78

Evaluate: [ \sqrt[3]{1 - \sqrt{3}} \cdot \sqrt[6]{4 + 2\sqrt{3}} ] This problem requires simplifying the radical expressions and finding the real value. We would analyze the expressions inside each root and attempt to rewrite in a simplified form.

Problem 79

Evaluate: $1^{-0.43} - 0.008^{-\frac{1}{3}} + (15,1)^0$ In this problem:

• $1^{-0.43}$ will be simplified, knowing that any power of $1$ is $1$.
• $0.008^{-\frac{1}{3}}$ simplifies by first converting $0.008$ to a fraction or a power of $10$.
• $(15,1)^0$ simplifies to $1$ since any number to the power $0$ is $1$.

Problem 80

Given: $x = \sqrt[8]{\frac{32\sqrt{2}}{\sqrt{8}}}$ Determine which of the following expressions results in an integer value. This involves simplifying the expression for $x$ and examining the powers to identify if any of the provided options result in an integer.

Problem 81

Simplify the expression: $\left( \left(a^{-\frac{3}{2}} b\right) (ab^{-2})^{-\frac{1}{2}} (a^{-1})^{-\frac{2}{3}} \right)^3$ This requires expanding each component within the parentheses using exponent rules, then simplifying by combining like terms.

Would you like a detailed solution for each problem?

Here are some related questions for practice:

1. Can you simplify $\sqrt[4]{81} \cdot \sqrt[3]{27}$?
2. How do you handle fractional exponents when simplifying expressions?
3. What is the effect of negative exponents on an expression?
4. How do you recognize if an expression with a square root or higher root results in an integer?
5. Can you rewrite $\sqrt[5]{32}$ in terms of fractional exponents?

Tip: When simplifying radicals and roots, look for patterns like perfect squares or cubes to make the process easier.