Name: ______Powers & Order of OperationsDecember 5, 2011

  1. What is the value of (185 + 278 + 579) – (85 + 178 + 279) ?
  1. Evaluate -2 + 7 ×–
  1. What number multiplied by itself is equal to the product of 6 and 150?
  1. A cell takes half a day to divide, forming two cells. If a cell culture starts with one cell, how many cells will there be after nine days?
  1. Find the value of the sum:

+ + + + … +

  1. What whole number between 100 and 200 is both a perfect square and a multiple of 7?
  1. Evaluate the following expression: . Express your answer as a reduced fraction.
  1. means that 2 is used as a factor n times. For example, means 2 x 2 x 2 x 2 = 16. We say that ends in 6 and that has a units digit of 6. If is multiplied out, what is the units (ones) digit of the product?

BONUS PROBLEMS

  1. Maria writes the same whole number in each box below and gets a true statement. Jon does the same as Maria and also gets a true statement. But Jon and Maria choose different numbers. What two numbers do they choose?

12 + ( □ x □ ) - (7 x □ ) = 0

  1. If x is multiplied out, what is the leading digit of the product? (Note: the leading digit of a number is the digit in the leftmost position.)
  1. Express the difference 10112 – 1102 in base 2.
  1. What is 20117 expressed as a base nine number?

SOLUTIONS:

  1. 500

To do this quickly in your head, note that each number in the first set is greater than its counterpart in the second set by an obvious amount:

185 – 85 = 100

278 – 178 = 100

579 – 279 = 300

Then 100 + 100 + 300 = 500

Or, you could simply do the full arithmetic, evaluating the expressions inside the parentheses first:

(185 + 278 + 579) – (85 + 178 + 279) =

1042 – 542 =

500

  1. 12

Evaluate the expression in the following order:

Please Excuse My Dear Aunt Sally

Parentheses Exponents Multiplication Division Addition Subtraction

-2 + 7 × 32 – (4 + 3)2 =

-2 + 7 × 32 – (7)2 =

-2 + 7 × 9 – 49 =

-2 + 63 – 49 =

61 – 49 =

12

  1. 30

6 x 150 = 900

Since 9 = 3 x 3, then

900 = 30 x 30

  1. 262,144

The cells double in count every half a day for nine days, so there will be 18 doublings. So starting with a single cell, that’s:

1 × 2 × 2 × 2 × 2 × ….. (total of 18 times)

which is:

1 × 218

or just:

218

Using the [xy]or [^] function on your calculator, this is 262,144.

  1. 50

The value of 1 raised to any integral power is still 1. For example, = 1 x 1 x 1 x 1 = 1. Since there are only even powers of 1, there are 50 addends of 1 to be summed. Therefore the sum is 50 x 1 = 50.

  1. 196

List the perfect squares between 100 and 200:

= 121

= 144

= 169

= 196

Of these, only 196 is a multiple of 7. The desired number is 196.

=

25 – 16 =

9

= 3

=

144 + 25 =

169

= 13

So, the answer is 3/13.

  1. ends in 2

Look for a pattern.

is 2
is 4
is 8
is 16, which ends in 6 / ends in 2
ends in 4
ends in 8
ends in 6 / ends in 2
ends in 4
ends in 8
ends in 6 / ends in 2
ends in 4
ends in 8
ends in 6

The unit’s digit repeats in groups of four: 2, 4, 8, 6. So , , , , and every power of 2 whose exponent is a multiple of four has a units digit of 6. Therefore has a units digit of 6 and has a units digit of 2.

  1. The two numbers are 3 and 4

Decide what values are possible and then substitute. To produce an end result of 0, 12 + ( □ x □ ) must be equal to (7 x □ ). Therefore ( □ x □ ) is less than (7 x □ ), and □is less than 7. Thus □ represents the whole numbers 0, 1, 2, 3, 4, 5, or 6. Replace all three boxes with the same number until two different choices work.

  1. 4

Noting that can be written as x , then x = x x . x is 1 followed by a string of 18 zeroes (), then the leading digit is 1 x = 4.

  1. 1012

10112

-1102

______

1012

Subtracting from right to left, in the ones, twos, fours and eights columns: 1 – 0 = 1; 1 – 1 = 0, and in base 2, 10 – 1 = 1 (1 eight – 1 four = 1 four).

  1. 851

20117 is (2 x 343) + (0 x 49) + (1 x 7) + 1 = 694 in base 10. Converting to base 9, 69410 = (8 x 81) + (5 x 9) + 1.

Cascade Ridge PTSA Math Club1Show your work on a separate sheet of paper.