5 Written Questions
5 Matching Questions
- Exponent Rules
- "Fewer Factors, More Multiples"
- Consecutive Multiples
- Multiplying & Dividing Signed Numbers
- a Factors divide into an integer "X" and are therefore less than or equal to X. Positive multiples multiply out from X and are therefore greater than or equal to X.
- all integers have a limited number of factors
- all integers have an infinite number of multiples
- b 1. Adding: when multiplying two numbers with the same base, combine exponents by adding
2. Subtracting: when dividing two numbers with the same base, combine exponents by subtracting them
3. Multiplying: when raising a power to a power, combine exponents by multiplying
4. Dividing: a negative exponent means putting the term with the exponent as the bottom of a fraction and making the exponent positive
- when there is a negative exponent, think reciprocal
5. Exponent of 1: any number that doesn't have an exponent implicitly has an exponent of 1
6. Exponent of 0: any non-zero base raised to the power of 0 is equal to 1
7. Fractional exponents: the numerator tells us the POWER to raise the base to, and the denominator tells us which ROOT to take
- c Whole numbers that are either positive or negative, including zero.
- adding, subtracting and multiplying result in integers.
- result of division is called a quotient.
- "divisible" = no remainders
- "divisor" or "factor" is a number that can evenly be divided into another
- d If Signs are the Same, the answer is poSitive but if Not, the answer is Negative.
- when multiplying or dividing a group of non-zero numbers, the result will be positive if you have an EVEN number of negative numbers.
- the result will be negative if you have an ODD number of negative numbers
- e Special cases of evenly spaced sets: all values in the set are multiples of the increment
- ex: 12,16,20,24 - increase by 4s, ea. element a multiple of 4
- these sets must be composed of integers
5 Multiple Choice Questions
- 1. Arithmetic mean = median.
- if there are an even number of elements, the median is the average of the two middle elements
2. Mean & median = average of the first and last element.
3. Sum of elements = mean * number of items in the set.
- The product of K consecutive integers is always divisible by K factorial (K!)
- ex: 3! = 3x2x1 = 6, always divisible by 3&2
- When you encounter any exponential expressions in which two or more terms in include something common in the base, consider factoring.
When an expression is given in factored form, consider distributing it.
- study book 1, pg. 164
- 1. N! Is the product of all positive integers smaller than or equal to N.
2. Works for multiples of two numbers if they share factors.
3. Any smaller factorial divides into any larger factorial - the smaller factor cancels completely.
- Get rid of square roots in the denominator of a simple expression by multiplying the numerator and denominator by that square root.
For complex expressions, use the conjugate of the denominator.
- change the sign of the square root term
5 True/False Questions
Multiple → Integer formed by multiplying that integer by any integer.
- negative multiples are possible
- zero is a multiple of every number
- every integer is a multiple of itself
Simplifying Roots 1 → Using prime factorization:
1. Factor the number under the radical sign into primes.
2. Pull out any pair of matching primes from under the radical sign, and place one of those primes outside the root.
3. Consolidate the expression.
When to use Prime Factorization → 1. Determine if x is divisible by y
2. Determine the greatest common factor of two numbers
3. Reducing fractions
4. Finding the least common multiple of a set of numbers
5. simplifying square roots
6. Determine the exponent of one side of an equation with integer constraints
How to Find GCF & LCM with Prime Columns → 1. Calculate the prime factors of each integer.
2. Create a column for each prime factor found within any of the integers.
3. Create a row for each integer.
4. In each cell, place the prime factor raised to a power. This power counts how many copies of the column's prime factor appear in the prime box of that row's integer.
Remainders 1 → On simple problems, pick numbers.
- add the desired remainder to a multiple of the divisor
- ex: need a number that leaves a remainder of 4 after dividing by 7; (7*2) + 4 = 18