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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.

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