Let's go through each part of the question systematically.

Given:

1. $a^{4} - a^{4} + 1 = 0$
2. $x = 2$, $y = 3$, and $z = 5$

Solution for Each Part:

(a) Simplify the Expression:

The expression to simplify is: $4a^2 + \frac{1}{4a^2} - 2 + 4a - \frac{1}{a}$

For simplification, substitute appropriate values for $a$ if possible or use algebraic identities. I'll assume further instruction is needed to solve this part.

(b) Determine the Value of $a^5 + \frac{1}{a^5}$

This requires finding $a^5 + \frac{1}{a^5}$. Normally, if values for $a$ are available or relations among powers of $a$, we can proceed with simplifications based on identities.

(c) Prove the Logarithmic Expression:

The expression to prove is: [ \log \sqrt[3]{y^3} + \log \sqrt{x^3} - \log \sqrt[3]{x^3 z^3} = \frac{3}{2} \log 1.2 ]

Using properties of logarithms, such as:

1. $\log \sqrt[n]{a} = \frac{1}{n} \log a$
2. $\log a + \log b = \log(ab)$
3. $\log a - \log b = \log \frac{a}{b}$

we can substitute the values $x = 2$, $y = 3$, and $z = 5$ to simplify and prove the expression.

Would you like detailed calculations for each part?

Related Questions

1. How do you approach simplifying logarithmic expressions with multiple terms?
2. What methods can help in solving power equations like $a^5 + \frac{1}{a^5}$?
3. How do properties of exponents assist in simplifying algebraic expressions?
4. What is the impact of substituting known values in complex algebraic simplifications?
5. Why is it important to follow order of operations in logarithmic and algebraic expressions?

Tip

Remember to use the properties of exponents and logarithms step-by-step to avoid mistakes, especially when working with nested or fractional powers.