Explanation

Let's find the sum of all the cube roots of unity.

The cube roots of unity are the solutions to the equation $z^3=1$.

These roots are:

1. $1$ (the real root)

2. $\omega =-\frac{1}{2}+i\frac{\sqrt{3}}{2}$

3. ${\omega }^2=-\frac{1}{2}-i\frac{\sqrt{3}}{2}$

Now, sum them:

$$\mathrm{<span\; style="font-weight:\; bolder;"><font\; color="\#0000ff">1</font></span>}<span\; style="font-weight:\; bolder;"><font\; color="\#0000ff">+</font></span>\mathrm{<span\; style="font-weight:\; bolder;"><font\; color="\#0000ff">\omega </font></span>}<span\; style="font-weight:\; bolder;"><font\; color="\#0000ff">+</font></span>{\mathrm{<span\; style="font-weight:\; bolder;"><font\; color="\#0000ff">\omega </font></span>}}^{\mathrm{<span\; style="font-weight:\; bolder;"><font\; color="\#0000ff">2</font></span>}}$$

It is a well-known property that the sum of the cube roots of unity is zero:

$$\mathrm{<span\; style="font-weight:\; bolder;"><font\; color="\#0000ff">1</font></span>}<span\; style="font-weight:\; bolder;"><font\; color="\#0000ff">+</font></span>\mathrm{<span\; style="font-weight:\; bolder;"><font\; color="\#0000ff">\omega </font></span>}<span\; style="font-weight:\; bolder;"><font\; color="\#0000ff">+</font></span>{\mathrm{<span\; style="font-weight:\; bolder;"><font\; color="\#0000ff">\omega </font></span>}}^{\mathrm{<span\; style="font-weight:\; bolder;"><font\; color="\#0000ff">2</font></span>}}<span\; style="font-weight:\; bolder;"><font\; color="\#0000ff">=</font></span>\mathrm{<span\; style="font-weight:\; bolder;"><font\; color="\#0000ff">0</font></span>}$$

Explanation

Let's simplify $\sqrt{-98}$ step by step.

1. Express under the square root as a product of a positive number and $-1$:

   $$\sqrt{<b><font\; color="\#0000ff">-</font></b>\mathrm{<b><font\; color="\#0000ff">98</font></b>}}<b><font\; color="\#0000ff">=</font></b>\sqrt{\mathrm{<b><font\; color="\#0000ff">98</font></b>}<b><font\; color="\#0000ff">\cdot </font></b><b><font\; color="\#0000ff">(</font></b><b><font\; color="\#0000ff">-</font></b>\mathrm{<b><font\; color="\#0000ff">1</font></b>}<b><font\; color="\#0000ff">)</font></b>}<b><font\; color="\#0000ff">=</font></b>\sqrt{\mathrm{<b><font\; color="\#0000ff">98</font></b>}}<b><font\; color="\#0000ff">\cdot </font></b>\sqrt{<b><font\; color="\#0000ff">-</font></b>\mathrm{<b><font\; color="\#0000ff">1</font></b>}}$$

2. Simplify $\sqrt{98}$:
   Factor 98:
   $98=49\times 2=7^2\times 2$
   So,

   $$\sqrt{\mathrm{<b><font\; color="\#0000ff">98</font></b>}}<b><font\; color="\#0000ff">=</font></b>\sqrt{\mathrm{<b><font\; color="\#0000ff">49</font></b>}<b><font\; color="\#0000ff">\times </font></b>\mathrm{<b><font\; color="\#0000ff">2</font></b>}}<b><font\; color="\#0000ff">=</font></b>\sqrt{\mathrm{<b><font\; color="\#0000ff">49</font></b>}}<b><font\; color="\#0000ff">\cdot </font></b>\sqrt{\mathrm{<b><font\; color="\#0000ff">2</font></b>}}<b><font\; color="\#0000ff">=</font></b>\mathrm{<b><font\; color="\#0000ff">7</font></b>}\sqrt{\mathrm{<b><font\; color="\#0000ff">2</font></b>}}$$

3. Recall that $\sqrt{-1}=i$:
   So,

   $$\sqrt{<b><font\; color="\#0000ff">-</font></b>\mathrm{<b><font\; color="\#0000ff">98</font></b>}}<b><font\; color="\#0000ff">=</font></b>\mathrm{<b><font\; color="\#0000ff">7</font></b>}\sqrt{\mathrm{<b><font\; color="\#0000ff">2</font></b>}}<b><font\; color="\#0000ff">\cdot </font></b>\mathrm{<b><font\; color="\#0000ff">i</font></b>}<b><font\; color="\#0000ff">=</font></b>\mathrm{<b><font\; color="\#0000ff">7</font></b>}\sqrt{\mathrm{<b><font\; color="\#0000ff">2</font></b>}}\mathrm{<b><font\; color="\#0000ff">i</font></b>}$$

Explanation

Let's solve the problem step by step.

Given:

• $z=i$

• We need to find $(\bar{z}{)}^2$, where $\bar{z}$ is the complex conjugate of $z$.

Step 1: Find the conjugate of $z$.
Since $z=i$, its conjugate is $\bar{z}=-i$.

Step 2: Square the conjugate.
So, $(\bar{z}{)}^2=(-i{)}^2=(-i)\times (-i)=i^2$.

Recall that $i^2=-1$.

Therefore, $(\bar{z}{)}^2=-1$.

Explanation

A vector in which the initial point is not specified is called a free vector.

A free vector is defined only by its magnitude and direction.

It can be moved parallel to itself anywhere in space without changing its properties.

Explanation

• $i^{21}=i$ (since 21 mod 4 = 1)

• $i^{27}=-i$ (since 27 mod 4 = 3)

• $i^{30}=-1$ (since 30 mod 4 = 2)

• Adding: $i+(-i)+(-1)=-1$

Explanation

If the intersection of two sets A and B is equal to their union, then A = B. 

Explanation

Step 1: Factor the equation

$2x^2-6x=0$

Factor out

$2x$

from both terms to get:

$2x(x-3)=0$

Step 2: Solve for x
Set each factor equal to zero to find the solutions:

$2x=0$

leads to

$x=0$

$x-3=0$

leads to

$x=3$

Step 3: Check the options
The solutions are

$x=0$

and

$x=3$

. The notationx∈W is {0, 3}means that the solutions are 0 and 3, and they are elements of the set of whole numbers (W), which is true. 

Explanation

For a differentiable function f(x), a relative maximum or minimum can only occur at a critical point. 

A critical point c is a value in the domain of the function where the first derivative (f'(c) is either zero or undefined. 

At a relative maximum, the slope of the tangent line to the graph of the function is zero, which is represented by the first derivative. 

This means that the function stops increasing and begins decreasing. 

Explanation

A - B (set difference) is the set of elements that are in A but not in B. So, A - B = { x | x ∈ A and x ∉ B }.

Since A and B are disjoint (no common elements), every element of A is not in B. Therefore, A - B = A.

So, (A - B) = A.

Now, we need (A - B) ∩ B = A ∩ B.

But since A and B are disjoint, A ∩ B = ∅.

Therefore, (A - B) ∩ B = ∅.

Explanation

The phrase "believe in" is the standard and correct phrasal verb to express faith or confidence in something or someone.

The sentence "She always believes in her own abilities" correctly conveys her confidence and trust in her capabilities.