CAHSEE practice: Problems 1-3

Problem number 1. Which number has the greatest absolute value? Absolute value is just the distance that the number is from 0, or how large the number is, regardless of its sign or its direction. So if you kind of visualize it on a number line, the absolute value of minus 17. Minus 17 is 17 to the left of 0. So its absolute value is just 17. The absolute value of minus 13? Well, it's 13 to the left of 0, if you imagine it on a number line. So it's just going to be 13. Absolute value of 15? Well, that's just going to be 15. It's 15 to the right of 0 on the number line. I think you see a pattern here. When you take the absolute value of a number, you essentially just disregard the sign. The absolute value of a negative number is going to be positive, and the absolute value of a positive number is going to be positive. The absolute value is always going to be positive, because you're just measuring how far you are away from 0 on the number line. And then finally, the absolute value of 19 is going to be equal to 19. So the number that has the greatest absolute value? Well, the greatest absolute value is right there, it's 19. So 19 has the greatest absolute value. So it also happens to be the largest number here. Just like that. Problem number 2. Maybe I'll do this in gray. Between which two integers is the value of the square root of 61? So the way you think about this is that you know there's some number-- the square root of 61 is going to be some decimal number, I don't know what it is-- but there's going to be some number that if I square it, is going to be a little bit lower than 61. And then some other integer, if I square it, should be a little bit greater than 61. So let's see if we can figure this out. So 6 squared is going to be 36. And then 7 squared is going to be 49. That number is not dark enough. Do a darker color. So this isn't going to be the case, right? Because the square of 61-- or 61 is greater than 49-- so the square root of 61 is going to be greater than the square root of 49. Or actually let me write it this way. I could rewrite 6 and 7 as the square root of 36 and the square root of 49. I can rewrite 7 and 8 as the square root of 49 and the square root of 64, right? 7 squared is 49, so the square root of 49 is 7. 8 squared is 64, so the square root of 64 is going to be 8. So I just rewrote 7 and 8 as the square root of 49 and the square root of 64. Well, the square root of 61 is clearly going to be between these two numbers, right? It's going to be between the square root of 49 and the square root of 64. The 61 is between 49 and 64. So our answer is B. Square root of 61 is between these two numbers, but these two numbers are the same thing as 7 and 8. Next problem, number 3. Use the addition problems below to answer the question. All right, so they wrote some addition problems right there. We're dealing with fractions. Let's see where this goes. Based on this pattern, what is the sum of 1/2 plus 1/4 plus 1/8 plus 1/16, all the way to 1/1024? So let's see if we can discern some type of a pattern here. Maybe I'll do this in red just to keep things interesting. So when I took the first two terms, I got 3/4. When I took the next two terms, or not the next two terms. When I add another term to it. When I add the 1/8 term to this over here, I got 7/8. Then when I add a 1/16 to 7/8, I get a 15/16. So the general pattern here is that whenever I add it up, my denominator and my sum-- I guess you could call it that way-- my denominator is going to be the same denominator that I have right there. And then my numerator, if I just look at all of the pattern here, just turns out to be 1 less than that, right? So when I added 1/2 plus 1/4, I got 4 in the denominator and 3 in the numerator. When I had 1/2 plus 1/4 plus 1/8, 8 in the denominator and 1 less than that in the numerator. When I added it all the way to 1/16, I got 16 in the denominator and 1 less in the numerator. When I added it all the way to 32, I got 32 in the denominator and 1 less in the numerator. And this only worked when I'm adding successive powers of, I guess you could call it 1/2, right? This is 1/2, 1/2 squared, 1/2 to the third, 1/2 to the fourth. So based on this pattern, what is the sum if I were to go all the way to 1/1024? Well, based on this pattern, my denominator is going to be 1,024, and my numerator is going to be 1 less than 1,024. So it's going to be 1,023. And that is choice C right there.