Algebra Problems of the Day (Integrated Algebra Regents, August 2012)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

August 2012

Part II: Each correct answer will receive 2 credits. Partial credit is possible.


31. State the value of the expression (4.1 X 10²)(2.4 X 10³) / (1.5 X 10⁷) in scientific notation.

Answer:

Remember the rules for exponents. Having base 10 is no different from having base x.

You can regroup them if you want to. Use your calculator to solve the first part. Remember to put the final answer back into scientific notation with the coefficient between 1 and 9.9999...

(4.1 X 10²)(2.4 X 10³) / (1.5 X 10⁷)
(4.1)(2.4)/(1.5) X (10²)(10³) / (10⁷)
6.56 X 10^{2 + 3 - 7}
6.56 X 10^-2

I'm a little surprised that they didn't form the problem so that the coefficient was 10 or greater or less than one.

32. Express the product of (x + 2)/2 and (4x + 20)/(x² + 6x + 8) in simplest form.

Answer:

With problems like this, factor whatever you can, and then cancel out common factors in the numerator and denominator which form a multiplicative identity.
(x + 2) / 2 * (4x + 20) / (x² + 6x + 8)
(x + 2) / 2 * ( (4)(x + 5) ) / ( (x + 4)(x + 2) )
1 / 1 * ( (2)(x + 5) ) / ( (x + 4)(1) )
( (2)(x + 5) ) / (x + 4)

(2)(x + 5)
x + 4

Either (2)(x + 5) or 2x + 10 were acceptable in the numerator.

33. On the set of axes below, graph y = 3^x over the interval -1 < x < 2.

Answer:

It is very important that you ONLY graph the domain that they gave you. Do NOT go beyond it. Do NOT draw arrows indicating the directions the line is going in. You CAN pick whatever scale you want for the x-axis. If you wanted, say, to count two boxes for every 1, that's fine.

Your graph should look like this:

At x = -1, the point should be at about 1/3 of the distance between 0 and 1. Do the best you can. Do NOT make a point on the x-axis.

More to come. Comments and questions welcome.

More Regents problems.