NotThatSmart

Let's know that XZW will form a triangle

\(\angle ZXW  = 90^\circ \\ \angle XWZ = 60^\circ\)

Thus, \( \angle XZW =  180 - 90 - 60  =   30\)

30 degrees is our answer. 

Thanks! :)

In the problem, it wanted us to find \(P_1 P_2 + P_2 P_3 + P_3 P_4 + \dots + P_9 P_{10} + P_{10} P_1\)

Notice that this is literally just the perimeter of decagon \(P_1 P_2 P_3 \dotsb P_{10}\)

The length of one side of the decagon is

\( \text{radius}/2 ( -1 + \sqrt {5}) \)

Multiplying this by 10 and we know the perimeter. We get

\( 10 (1/2) ( -1 +\sqrt 5) = 5 ( -1 + \sqrt 5) ≈ 6.18\)

Thus, 6.18 is approximately our answer. 

Thanks! :)

3. I'm assuming we are trying to find the value of \(\frac{8^{x}}{2^{y}}\)

Let's cite some equations that could help us in these equations. There are two main ideas we will be applying for this problem. 

\(a^{bc}=(a^{b})^{c}\) \(\frac{a^{b}}{a^{c}}=a^{b-c}\)

Now, let's focus on what we are trying to find. Let's simplify it a bit to get a better shot at figuring out our final answer. 

\(\frac{8^{x}}{2^{y}}=\frac{(2^{3})^{x}}{2^{y}}\)

This is where we apply the second equation stated. This allows us to put x and y into one exponent. 

\(\frac{2^{3x}}{2^{y}}=2^{3x-y}\)

Wait! We recognize this 3x - y from somewhere! The value was given to us in the problem! It was stated that \(3x-y=12\)

Thus, we plug in this value, we have

\(2^{12} = 4096\)

Thus, our final answer is 4096. 

Thanks! :)

Look at the image!

2. I don't have much time to explain...hopefully you understand my work...sorry

First, put everything on the right side into one fraction

\(\frac{24x^{2}+25x-47}{ax-2}=\frac{(-8x-3)(ax-2)-53}{ax-2}\)

Simplify the numerator on right side. 

\(\frac{24x^{2}+25x-47}{ax-2}=\frac{-8ax^{2}+16x-3ax+6-53}{ax-2}\) \(\frac{24x^{2}+25x-47}{ax-2}=\frac{-8ax^{2}+(-3a+16)x-47}{ax-2}\)

Since both have the same denominator, their numerators must be equal. 

\(24x^{2}+25x-47=-8ax^{2}+(-3a+16)x-47\)

Since the coefficients must correspond, we can right the equation

\(24x^{2}=-8ax^{2}\) \(24=-8a\) \(a=-3\)

Thus, b is the right answer. 

Thanks! :)

ummm...I would give a solid solution...but this was answered VERY well here.

1. https://www.doubtnut.com/qna/181168330

answer is d, btw :)

There's many ways to do this problem, but I prefer using Euler's Totient Function, mainly because when facing larger and more complicated numbers, listing is not going to do the trick. 

So here's the Totient Function. It essentially states that

\(\frac{\text{number we start with}}{\text{product of the prime factors}} \cdot \text{product of every prime factor -1}\) and this is our final answer

Thus, we can apply this to 20. 

We have \(20 = 2^2 * 5\)

2 and 5 are the two prime factors, so we have

\(\frac{20}{2*5} \cdot (2-1)(5-1) = 2 * 4 = 8\)

Thus, 8 is the final answer. 

If you want to learn more, here's another question that revolves around the equation. 

Let's set some variables and make some observations. 

Let's let \(d = a_2 -a_1\)

Now, let's note something important. Note that

\(S_5 = 5a_1 + (d + 2d + 3d + 4d) = 5a_1 + 10d\\ S_{10} = 10a_1 + (d + 2d + \cdots + 9d) = 10a_1 + 45d\\ S_{15} = 15a_1 + (d + 2d + \cdots + 14d) = 15a_1 + 105d\)

These observations are really important for us to solve the problem. 

Let's take two of these cases. We can write a system of equations, and we get

\(\begin{cases} 5a_1 + 10d = \dfrac15\\ 10a_1 + 45d = \dfrac1{10} \end{cases}\)

Now, we solve for a1 and d. 

Solving this system of equations, we get

\(d = -\dfrac3{250}\\ a_1 = \dfrac8{125}\)

Plugging this into S15, we get \(S_{15} = 15a_1 + 105d = -\dfrac3{10}\)

So -3/10 is our final answer. 

Thanks! :)