Here are the solutions to your math assignment problems:

1. Express the complex number $\sqrt{ }$ in standard form.

There seems to be missing information. Could you clarify the full expression?

---

2. Use De Moivre’s theorem to compute $( a + bi )^n$ and write the result in standard form.

De Moivre’s theorem states that for a complex number in polar form $z = r (\cos \theta + i \sin \theta)$, its power is given by:

$z^n = r^n (\cos(n\theta) + i\sin(n\theta))$

If you provide the specific complex number and power, I can compute the result.

---

3. Find the cube roots of $\sqrt{ }$ and write each root in standard form.

I need the full expression to proceed. Please provide the exact complex number.

---

4. Graph the relation $f(x) = \text{some function}$ and determine its domain and range.

To graph the function and determine its domain and range, I need the full function expression. Could you provide the equation?

---

5. Let $f(x)$ and $g(x)$ be given. Find $(f \circ g)(x)$.

The function compositions depend on the given functions. Could you provide the full expressions for $f(x)$ and $g(x)$?

---

6. Given $f(x) =$ (a rational function), determine the following:

a) Find the domain of $f(x)$.

The domain consists of all real numbers except those that make the denominator zero.

b) Find the y-intercept of $f(x)$.

Set $x = 0$ and solve for $f(0)$.

c) Find the x-intercepts of $f(x)$.

Set $f(x) = 0$ and solve for $x$.

d) Find all asymptotes of $f(x)$.

• Vertical asymptotes: Solve where the denominator is zero.
• Horizontal asymptotes: Analyze the degree of the numerator and denominator.
• Oblique asymptotes: If the degree of the numerator is one higher than the denominator, perform polynomial division.

e) Draw the graph of $f(x)$.

After identifying key points and asymptotes, plot the function.

---

7. Solve the given equations:

a) Solve $( \text{some equation} )$.

Please provide the full equation for solving.

b) Solve $x^2 + 4x - 5 = 0$.

Solve using factoring, the quadratic formula, or completing the square.

c) Solve $\log_2(x + 1) = 3$.

Rewrite in exponential form:

$x + 1 = 2^3$

$x = 8 - 1 = 7$

---

Next Steps

• Can you clarify the missing expressions for questions 1, 2, 3, 4, and 5?
• Would you like a step-by-step explanation for any of these problems?

---

Related Questions

1. What is De Moivre’s theorem, and how is it used in complex numbers?
2. How do you determine the domain and range of a function from its equation?
3. What are the different types of asymptotes and how do you find them?
4. How do you solve logarithmic equations using exponentials?
5. What are the cube roots of unity, and how do they relate to complex numbers?

---

Tip:

For solving quadratic equations, always check for factoring first—it’s often the quickest method!