NotThatSmart

Now, let's note something really important. 

To determine the units digit of a number, we must focus on the unit digits of the two numbers we are multiplying. 

Any other term actually doesn't matter. 

Thus, we have

\(3*7=21\)

So the units digit is 1. 

Also, we could multiply these two numbers together, since both are small. 

\(13*287=3731\)

The units digit is 1. 

So our final answer is 1. 

Thanks! :)

Ok, we can go one by one to solve these problems. 

Hoever, let's note that if we have f(x) = x, then we set the right side of the function to equal x.

Then, we see if the x value satisfies the conditions given. 

First, we have \(f(x) = 2x + 8\). Setting this to equal x, we fuind that \(x=2x+8\)

Solving for x, we find that \(x=-8\). This doesn't satisfy the condition \(1 \le x \le 2\), so it is not a valid solution. 

Next, we have \(f(x) = 13 - 5x\). Setting this to equal x, we have \(x=13-5x\)

Finding the value of x, we get \(x=13/6\). This satisfies the inerval, so it is a solution. 

Third, we have \(f(x) = 20 - 14x\). Since this equals x, we can write \(x = 20 - 14x\) 

Trying to find x, we have that \(x=20/15=4/3\). This does NOT satisfy the interval, so it is not a solution. 

Lastly, we have \(f(x) = 40 + 5x \). setting this function to equal x, we have \(x=40+5x\)

Solving for x, we get \(x=-10\), which is invalid. 

Thus, the only value that works i 13/6. 

So our answer is 13/6. 

Thanks! :)

In order to solve this problem, let's analyze somethings about it. 

First, let's note that right triangles \(ABC, ABE, ABD, BHD, BEC\) are all 30-60-90 triangles. 

Also, note that right triangles \( ABE,ADC \) are 45-45-90 triangles. 

In order to solve the problem, let's draw a line to help find HCA for us. 

Let's draw the altitude of ABC by connecting c to line segment AB. 

This creates a right triangle!

Since we know that

\(\angle BAC =60^\circ\)

HCA would have to be 30 degrees because of the right triangle.

So we have

\(\angle HCA = 30^\circ\)

So 30 degrees is our answer. 

Thanks! :)

In order to know what the 43rd term is, we must know the common difference. 

Luckily, the problem basically tells us what the difference is. 

Let's let that difference be d. We have from the problem that

\(1/4 + 10d = -1/5 \\ 10d = -1/5 - 1/4 = -9/20 \\\ d = -9/200\)

Now, we biuld off of the 33rd term. Since we know it is 1/5, we simply just have to add the d a number of times. 

We have

\( -1/5 + 10d = -1/5 + 10 (-9/200) = -1/5 - 9/20 = -65/100 = -13 / 20\)

So our answer is -13/20

Thanks! :)

Let's first focus on the first equation. 

Squaring both sides of the first equation, we find that

\(a^2 + 2ab + b^2 = 16 \)

Reaaranging the second equation from the problem, we get that

\(a^2 -ab+ b^2 = 6\)

Now, moving on, let's note something important. We have that

\(a^3 + b^3 = (a + b) ( a^2 + b^2 - ab) \)

We already have all the terms needed to solve the problem. We know every single number. 

The first term parenthesis is the first equation, and the second parenthesis is the second equation. 

Thus, plugging in 6 and 4, we get that

\(a^3 + b^3 = 6*4 =24\)

So 24 is our answer. 

Thanks! :)

Let's note something for this problem first. 

We are basically taking a look at the squares of numbers, and eeing the largest 3 digit number. 

Thus, let's test some integers. We find that

\(31^2 = 961 \\ 32^2 = 1024 \)

This shows that 31 is the largest value squared. 

Thus, 961 is our answer. 

So we have \(x=961\)

Thanks! :)

Let's first multiply the terms together and see what we get. 

Multiplying \(x^3(x^2+tx)\), we use the distrbutive property. 

Factoring in x^3, we get that

\(x^5+tx^4\)

There is no x^2 term in this expansion of the product. 

This means t can pretty much be anything unless it contain x^-2. 

Other than that, t could be pretty much any number or polynomial. 

Thanks! :)

Now, there are two ways to do this problem. 

First off, let's look at the possibilities of how we can create x. 

We have x and 14, which gets us 14x. 

Next, we have -23x and 1, which gives us -23x. 

Thus, the coefficient is simply \(14x-23x = -9x\)

We could also just expand the entire thing. We find that after expanding out the entire thing, we have

\(x^{7}-7x^{6}+10x^{5}-13x^{4}+8x^{2}-9x+14\)

Thus, the coefficient of x is just \(-9\)

So -9 is our answer. 

Thanks! :)