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That simplifies to 2*3^(1/3).  Hope this helps!

For this problem, you must consider magnitude and argument of complex numbers geometrically.

The answers are as follows:

r_1 = 16

theta_1 = 3*pi/2

r_2 = sqrt(3)

theta_2 = 5*pi/4

r_3 = 12

theta_3 = pi/3

For this problem, you must consider magnitude and argument of complex numbers geometrically.

The answers are as follows:

r_1 = 16

theta_1 = 3*pi/2

r_2 = sqrt(3)

theta_2 = 5*pi/4

r_3 = 12

theta_3 = pi/3

For this problem, you must consider magnitude of complex numbers geometrically.

For the first part, the answers are |v| = 2, |w| = 3, |z| = 3.

For the second part, the answers are |v + w| = sqrt(2), |w + z| = 2, |z + v| = sqrt(3).

For this problem, you must consider magnitude of complex numbers geometrically.

For the first part, the answers are |v| = 2, |w| = 3, |z| = 3.

For the second part, the answers are |v + w| = sqrt(2), |w + z| = 2, |z + v| = sqrt(3).

For this problem, you must consider addition and subtraction of complex numbers geometrically.  The answer turns out to be s, t, u.

For this problem, you must consider addition and subtraction of complex numbers geometrically.  The answer turns out to be s, t, u.

P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1 is equal to 20.

The time difference between driving at 60 mph and 40 mph is 45 minutes, which is 3/4 of an hour. So, the equation for the time you spent driving is:

time = (rate * time_difference) / (rate - new_rate)

Plugging in the known values, we get:

time = (40 mph * 3/4 hours) / (40 mph - 25 mph)

time = 7.5 hours

Since the rate is in miles per hour and the time is in hours, the distance traveled is simply the product of the two:

distance = rate * time

distance = 40 mph * 7.5 hours

distance = 300 miles

Therefore, you drove 300 miles to the beach.

Where the graphs intersect, they are equal :

x^3 + x^2 -3x + 5   =  x^3 + 2x^2

x^2 -3x + 5 = 2x^2 

0 =  x^2 +3x -5

   Now use Quadratic Formula with   a = 1  b = 3  and c = -5    to find x = 1.193    and -4.193

   Now use these values of 'x' in either of the equations to find the corresponding 'y' coordinates.  

Let \( f(N) = (3N^2 + 8N - 3)(pN - 1) \). We need to find the remainder when \( f(N) \) is divided by \( N + 1 \). According to the Remainder Theorem, the remainder of \( f(N) \) when divided by \( N + 1 \) is equal to \( f(-1) \).

To find \( f(-1) \), we start by calculating \( 3(-1)^2 + 8(-1) - 3 \):

\[
3(-1)^2 + 8(-1) - 3 = 3(1) - 8 - 3 = 3 - 8 - 3 = -8
\]

Next, we calculate \( p(-1) - 1 \):

\[
p(-1) - 1 = -p - 1
\]

Now we can find \( f(-1) \):

\[
f(-1) = (-8)(-p - 1) = 8(p + 1)
\]

According to the problem, the remainder \( f(-1) \) is given to be 24:

\[
8(p + 1) = 24
\]

Dividing both sides by 8, we find:

\[
p + 1 = 3 \implies p = 2
\]

Now we want to find the remainder when \( p^4 \) is divided by 10. Calculating \( p^4 \):

\[
p^4 = 2^4 = 16
\]

Now we find the remainder of 16 when divided by 10:

\[
16 \mod 10 = 6
\]

Thus, the remainder when \( p^4 \) is divided by 10 is:

\[
\boxed{6}
\]

We can simplify the given equation: 5a + 3b + 5c = 4a + 5b + 4c a - 2b - c = 0 a = 2b + c

Now we can substitute a in the expression for N: N = 5(2b + c) + 3b + 5c N = 13b + 10c

We are given that 131 < N < 150. Let's try some values for b and c:

If b = 10 and c = 1, then N = 140, which is within the given range.

If b = 11 and c = 1, then N = 153, which is outside the given range.

Therefore, the maximum possible value of b is 10.

Now we can find the maximum possible value of a + 2b + 3c: a + 2b + 3c = (2b + c) + 2b + 3c = 5b + 4c

Substituting b = 10 and c = 1, we get: 5b + 4c = 5(10) + 4(1) = 54

Therefore, the maximum possible value of a + 2b + 3c is 54.

Start with 1011  then add 21 sequentially until you reach 1998 to find   48 numbers