Solution

1. 6

   Multiplying 0.02468 by a positive power of 10 will shift the decimal point to the right. Simply shift the decimal point to the right until the result is greater than 10,000. Keep track of how many times you shift the decimal point. Shifting the decimal point five times results in 2,468. This is still less than 10,000. Shifting one more place yields 24,680, which is greater than 10,000.

2. {−2, −3}

   To give 2002 a value between 1 and 100, you must shift the decimal point to change the number to 2.002 or 20.02. This requires a shift of either two or three places to the left. Remember that while multiplication shifts the decimal point to the right, division shifts it to the left. To shift the decimal point two places to the left, divide by 10². To shift it three places to the left, divide by 10³. Therefore, the exponent −b = {2, 3}, and b = {−2, −3}.

3. 90,000

   Use the Heavy Division Shortcut to estimate:

   
4. 5,200

   Use the order of operations, PEMDAS (Parentheses, Exponents, Multiplication & Division, Addition & Subtraction), to simplify. Remember that the numerator acts as a parentheses in a fraction:

   4.5 × 2 = 9

   
5. 0.0375

   First, rewrite the numbers in standard notation by shifting the decimal point. Then, add zeroes, line up the decimal points, and subtract:

      0.0400
   − 0.0025
       0.0375

6. 46

   To divide by a positive power of 10, shift the decimal point to the left. This yields 45.63021. To round to the nearest whole number, look at the tenths place. The digit in the tenths place, 6, is more than 5. Therefore, the number is closest to 46.

7. 0.016

   Use the order of operations, PEMDAS (Parentheses, Exponents, Multiplication & Division, Addition & Subtraction), to simplify. Shift the decimals in the numerator and denominator so that you are dividing by an integer:

   
8. 0.0004

   Use the order of operations, PEMDAS (Parentheses, Exponents, Multiplication & Division, Addition & Subtraction), to simplify:

   First, add 1.08 + 6.9 by lining up the decimal points.
   Then, subtract 7.98 from 8 by lining up the decimal points, adding zeroes to make the decimals the same length.
   Finally, square 0.02, by multiplying 2 × 2, and then recognizing that (0.02) × (0.02) has a total of four digits to the right of the decimal point. 4 → 0.0004

9. {−3, −4}

   To give −37,129 a value between −100 and −1, you must shift the decimal point to change the number to −37.129 or −3.7129. This requires a shift of either three or four places to the left. Remember that multiplication shifts the decimal point to the right. To shift the decimal point three places to the left, you would multiply by 10⁻³. To shift it four places to the left, you would multiply by 10⁻⁴. Therefore, the exponent j = {−3, −4}.

10. 0.009

    Shift the decimal point two spaces to eliminate the decimal point in the denominator:

    

    Then divide. First, drop the three decimal places: 81 ÷ 9 = 9. Then put the three decimal places back: 0.009.

11. a = 2, b = 2, c = 1

    

12. 8

    You can focus on the last digits only: 16⁴ × 27³ → 6⁴ × 7³

    6⁴ → 6² × 6² → 36 × 36 → 36 → 6
        7³ → 7² × 7 → 49 × 7 → 63 → 3
                                          6 × 3 = 18 → 8

13. (C)

    In Quantity A, when you divide by 10 raised to a negative power, move the decimal to the right, so that 573 becomes 57,300:

    Quantity A	Quantity B
    	0.573 × 105

    In Quantity B, when you multiply by 10 raised to a positive power, move the decimal to the right, so that 0.573 becomes 57,300.

    Quantity A	Quantity B
    57,300	0.573 × 105 = 57,300

    Therefore, the two quantities are equal.

14. (B)

    Quantity A looks pretty intimidating at first. The trap here is to try to find an exact value for the expression in Quantity A. Let’s estimate instead:

    603,789,420 ≈ 600,000,000.

    13.3 × 10⁷ ≈ 133,000,000, or even better, 130,000,000

    Quantity A	Quantity B
    	5

    Now you can cross off the zeros.

    Quantity A	Quantity B
    	5

    Multiply both quantities by 13:

    Quantity A	Quantity B
    	5 × 13 = 65

    Therefore, Quantity B is greater.

15. (C)

    Whenever you multiply fractions or decimals, it is usually preferable to convert the numbers to fractions. Simplify the parentheses in Quantity A and convert 0.25 to a fraction:

    Quantity A	Quantity B
    	0.35

    Now compare 7/20 and 0.35. Put 0.35 into fraction form and reduce:

    Quantity A	Quantity B

    Therefore, the quantities are the same.