Maintaining Mathematical Proficiency

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Exercises 1 The first step in simplifying this expression is to identify which, if any, terms can be combined. Remember, only like terms — constant terms or terms with the same variable — can be combined. 3x−7+2x​ In this case, we have two x-terms and a constant. Only the x-terms can be combined, so to simplify the expression we will rearrange it according to the Commutative Property of Addition and then evaluate the sum of the like terms. 3x−7+2xCommutative Property of Addition3x+2x−7Add terms5x−7
Exercises 2 The first step in simplifying this expression is to identify which, if any, terms can be combined. Remember, only like terms — constant terms or terms with the same variable — can be combined. 4r+6−9r−1​ In this case, we have two r-terms and two constants. Both the x-termsand constants can be combined, so to simplify the expression we will rearrange them according to the Commutative Property of Addition and then evaluate the difference between the like terms. 4r+6−9r−1Commutative Property of Addition4r−9r+6−1Subtract terms-5r+5
Exercises 3 The first step in simplifying this expression is to identify which, if any, terms can be combined. Remember, only like terms — constant terms or terms with the same variable — can be combined. -5t+3−t−4+8t​ In this case, we have three t-terms and two constants. Both the x-terms and constants can be combined, so to simplify the expression we will rearrange them according to the Commutative Property of Addition and then evaluate the difference between the like terms. -5t+3−t−4+8tCommutative Property of Addition8t−5t−t+3−4Subtract terms2t−1
Exercises 4 We will begin simplifying by distributing the 3 to the terms inside the parentheses. 3(s−1)+5Distribute 33s−3+5Add terms3s+2 Since 3s and 2 are not like terms, the expression cannot be simplified further.
Exercises 5 We will begin simplifying by distributing the -7 to the terms inside the parentheses. 2m−7(3−m)Distribute -72m−21−7(-m)-a(-b)=a⋅b2m−21+7mCommutative Property of Multiplication2m+7m−21Add terms9m−21 Since 9m and -21 are not like terms, the expression cannot be simplified further.
Exercises 6 We will begin simplifying by distributing the 4 and then -1 to the terms inside the parentheses. 4(h+6)−(h−2)Distribute 44h+24−(h−2)Distribute -14h+24−h+2Commutative Property of Multiplication4h−h+24+2Subtract term3h+24+2Add terms3h+26 Since 3h and 26 are not like terms, the expression cannot be simplified further.
Exercises 7 To determine the greatest common factor between a set of numbers, let's first break down each of the given numbers into their prime factors. We'll also take note of all prime factors that are common between each of the given numbers.Given numberPrime FactorizationCommon Prime Factors 202⋅2⋅52, 2 362⋅2⋅3⋅32, 2 For the given values, the common prime factors are: 2, and 2. Calculating the product of these factors will give us the GCF of the given set. 2⋅2=4​
Exercises 8 To determine the greatest common factor between a set of numbers, let's first break down each of the given numbers into their prime factors. We'll also take note of all prime factors that are common between each of the given numbers.Given numberPrime FactorizationCommon Prime Factors 422⋅3⋅73 633⋅3⋅73 For the given values, the common prime factor is: 3. So 3 is the GCF of the given set.
Exercises 9 To determine the greatest common factor between a set of numbers, let's first break down each of the given numbers into their prime factors. We'll also take note of all prime factors that are common between each of the given numbers.Given numberPrime FactorizationCommon Prime Factors 542⋅3⋅3⋅33, 3, 3 813⋅3⋅3⋅33, 3, 3 For the given values, the common prime factors are: 3, 3 and 3. Calculating the product of these factors will give us the GCF of the given set. 3⋅3⋅3=27​
Exercises 10 To determine the greatest common factor between a set of numbers, let's first break down each of the given numbers into their prime factors. We'll also take note of all prime factors that are common between each of the given numbers.Given numberPrime FactorizationCommon Prime Factors 722⋅2⋅2⋅3⋅32, 2, 3 842⋅2⋅3⋅72, 2, 3 For the given values, the common prime factors are: 2, 2 and 3. Calculating the product of these factors will give us the GCF of the given set. 2⋅2⋅3=12​
Exercises 11 To determine the greatest common factor between a set of numbers, let's first break down each of the given numbers into their prime factors. We'll also take note of all prime factors that are common between each of the given numbers.Given numberPrime FactorizationCommon Prime Factors 282⋅2⋅72, 2 642⋅2⋅2⋅2⋅2⋅22, 2 For the given values, the common prime factors are: 2 and 2. Calculating the product of these factors will give us the GCF of the given set. 2⋅2=4​
Exercises 12 To determine the greatest common factor between a set of numbers, let's first break down each of the given numbers into their prime factors. We'll also take note of all prime factors that are common between each of the given numbers.Given numberPrime FactorizationCommon Prime Factors 302⋅3⋅5- 777⋅11- For the given values, there are no common prime factors. Therefore, there is no GCF of the given set.
Exercises 13 Let's star by reviewing what prime numbers are. Prime numbers are only divisible by 1 and themselves. For this reason, different prime numbers cannot have common factors. 13=1⋅1311=1⋅11​ Prime numbers are a special case. However, these are not the only examples of integer numbers without common factors. We can construct pairs of numbers with this characteristic by multiplying different prime numbers. 2⋅7⋅13=1823⋅5⋅11=165​ Numbers like 182 and 165, which have no common factors, are called relatively prime numbers or coprime numbers.