1.      D     

This question tests your understanding of order of operations, exponents, and logarithms.

We’ll solve it step by step. The numerator simplifies to

The denominator is 5, because 2⁵ = 32. So the answer is ⁴⁰⁄₅ = 8.

2.      E     

The product of two negative numbers must be positive, and the product of an even number and any other number is even: therefore, (a b) must be even and positive. The square of an even number is even and positive, so c² is even and positive. Since two positive even numbers will sum to a positive even number, (a b) + c² must be positive and even. When you add one to this value, the end result is odd and positive. 9 is the only answer choice that is odd and positive.

3.      B     

Find the prime factorizations of 24 and 42 to find their LCM and GCF.

The LCM of the two numbers is 2³ 3 7 = 168, and the GCF of the two numbers is 2 3 = 6. Now you just need to find the absolute value of the difference between the LCM and the GCF: |168 – 6| = |162| = 162.

4.      C     

Fractions are equivalent to each other if the numerator and denominator of one fraction can be multiplied by the same scalar, and the result is the other fraction. After reducing the first two fractions, you should have realized that they were both equivalent to ³/₇. As soon as you got to the third one and found that it was already in reduced form and not equal to ³ /₇, you could have stopped. If you checked the last two, you found that they, too, are equal to ³/₇.

5.      B     

To answer this problem, use scientific notation. When you type 5³³ into your calculator, it is approximately 1.16 10²³. This means that the decimal point has been moved over 23 decimal places, so there must be 23 + 1 = 24 digits in the full number.