Park Forest Math Team. Meet #3. Self-study Packet

Transcription

1 Park Forest Math Team Meet # Self-study Packet Problem Categories for this Meet (in addition to topics of earlier meets): 1. Mystery: Problem solving 2. Geometry: Properties of Polygons, Pythagorean Theorem. Number Theory: Bases, Scientific Notation 4. : Integral Powers (positive, negative, and zero), roots up to the sixth 5. Algebra: Absolute Value, Inequalities in one variable including interpreting line graphs

2 Important information you need to know about ARITHMETIC Integral Powers (Positive, Negative, and Zero), Roots up to the Sixth Anything to the power of zero is 1. A number to a negative power is the same thing as the reciprocal of that number to the opposite positive power. When multiplying two powers with the same base, keep the base and add the exponents. When dividing two powers with the same base, keep the base and subtract the exponents. Laws of Exponents Common Powers worth Memorizing (Focus on learning the squares and cubes and the higher powers of the numbers less than six) , , , , , , ,049 51, , ,000 1,000,000 Squares of higher numbers: =121, =144, =169, =196, =225, =256, =289, =24, =61, =400

3 Category 4 Meet # - January, ) Find the value of 2) Evaluate ) means or, equivalently,. Find the value of On this date, January 8, Charles Babbage patented the first computer in the year ANSWERS 1) 2) )

4 Solutions to Category 4 Meet # - January, ) 425 2) 16 ) 872 1) 2) ) (Multiply before subtracting!)

5 Category 4 Meet #, January Evaluate the following expression. You should get a whole number Evaluate the expression below for a = 18 and b = 24. Express your answer as a mixed number in lowest terms. a b a b How many whole numbers are there between 1000 and 1000? whole numbers

6 Solutions to Category 4 Meet #, January = = = = whole numbers 2. Notice that the first term in this expression is raised to the zero power, so it s value is 1. As for the second term, we substitute 18 for a and 24 for b, and evaluate as follows: a b 2 1 = = 24 2 = = = = 81 8 =101 8 Putting the two terms together, we get a b a b 2 1 = = We need to find a perfect fourth power that is slightly less than 1000 and a perfect square that is slightly greater than The two short 4 lists below can help. From the list on the left, we see that 1000 must be greater than 5, but less than 6. From the list on the right, we see that 1000 must be greater than 1, but less than 2. There are 1 whole numbers less than 2, but we need to exclude 5 of these, so there are 1 5 = 26 whole numbers between 5 and = = = = = = 1024

7 Meet # January 2011 Category 4 1. Evaluate the following expression: 2. is a natural number that makes the equation below true. Find the value of.. is a natural number greater than 1 such that the difference between the 4 th and rd powers of is times the difference between the rd and 2 nd powers. What is the value of?

8 Meet # January 2011 Solutions to Category We know that. Taking the common factors out in each side of the equation we get: so we can divide both sides by to get. but accepted answer since 0 = 4 times 0. Also accepted was 1,4.

9 Category 4 Meet #, January What is the largest possible integer value of if? 2. What is the value of the expression below? Express your answer as a common fraction.. There are exactly 8 whole numbers that are greater than and less than. If is a whole number, what is the largest possible value of?

10 Solutions to Category 4 Meet #, January = The largest power of less than that is 8 = 6561, so a = and, so is between 7 and 8. The first whole number greater than then is 8. The 8 th whole number greater than is 45. Therefore must be larger than 45, but not larger than 46 or there would be 9 numbers between and = 2025 while 46 2 = So the largest whole number value of would be 2116 as 45 is less than while 46 is not.

11 Category 4 Meet #, January Evaluate the following expression. Write your result as a mixed number in lowest terms What is the sum of all the integer values that are greater than and less than 2 50?. If n is a positive integer less than 160,000, what value of n will make the expression below a whole number? n

12 Solutions to Category 4 Meet #, January We can evaluate the expression as follows: = = = Since 4 5 = 1024 and 5 5 = 125, we know that 5 is the least 5 integer greater than Since 7 2 = 49 and 8 2 = 64, we 2 know that 7 is the greatest integer less than 50. Thus the sum 5 of all the integer values that are greater than 2000 and less 2 than 50 is = 18.. Let s simplify the expression first n ( 7) n n We are looking for a perfect cube under the radical. The perfect cubes greater than 7 are 8, 27, 64, etc. If we want to 4 make 8, then n would have to be 1, which means n 4 = 1 4 = 1. 4 If we want to make 27, then n would have to be 20, which means n 4 = 20 4 = 160,000. But n must be less than 160,000, so n must be 1.

13 Category 4 Meet #, January Evaluate the expression 4 xy if x = 24 and y = Evaluate the expression below. Write your result as a simplified fraction How many whole numbers are there between 25 and 2005?

14 Solutions to Category 4 Average team got points, or 1.0 questions correct The best team score was 2 out of 6 points. Meet #, January The expression can be evaluated as follows: 4 xy = = 4 ( ) ( 2 ) = = 2 = The expression can be evaluated as follows: = = 2 2 = 2 ( ) 2 ( 2 ) = = = 1 6. The value of 25 is greater than 2 but less than, since 2 = 8 and = 27. Similarly, the value of 2005 is greater than 12 but less than 1, since 12 = 1728 and 1 = Thus, the whole numbers between 25 and 2005 are through 12, which is ten (10) numbers.