Question

3-5 hw multiplying & dividing integer practice finding the product and quotient of the following problems using the appropriate integer rules. 1. $4(-3) =$ 2. $\frac{18}{-3} =$ 3. $(-9)(-8) =$ 4. $-42 div -7 =$ 5. $2 cdot -3 cdot -4 =$ 6. $-15 div -3 cdot -6 =$ 7. when multiplying 2 integers and they are both negative, your answer is ____. 8. when diving 2 integers and 1 of them is negative, your answer is __. 9. when multiplying with integers, a negative times a positive or a positive times a negative, the product will be __. 10. multiplication/division: different signs are __. positive negative 11. multiplication/division: same signs are __. positive negative 12. when multiplying 3 integers and 2 of them are negative, your answer will be ____.

Explanation:

Step1: Solve \( 4(-3) \)

Using the rule for multiplying a positive and a negative integer: \( a \times (-b) = -ab \) (where \( a = 4 \), \( b = 3 \)). So \( 4\times(-3)=-12 \).

Step2: Solve \( \frac{18}{-3} \)

Using the rule for dividing a positive by a negative integer: \( \frac{a}{-b}=-\frac{a}{b} \) (where \( a = 18 \), \( b = 3 \)). So \( \frac{18}{-3}=-6 \).

Step3: Solve \( (-9)(-8) \)

Using the rule for multiplying two negative integers: \( (-a)\times(-b)=ab \) (where \( a = 9 \), \( b = 8 \)). So \( (-9)\times(-8)=72 \).

Step4: Solve \( -42\div -7 \)

Using the rule for dividing two negative integers: \( \frac{-a}{-b}=\frac{a}{b} \) (where \( a = 42 \), \( b = 7 \)). So \( -42\div -7 = 6 \).

Step5: Solve \( 2\cdot -3\cdot -4 \)

First, multiply \( 2 \) and \( -3 \): \( 2\times(-3)=-6 \) (positive times negative is negative). Then multiply \( -6 \) and \( -4 \): \( (-6)\times(-4)=24 \) (negative times negative is positive).

Step6: Solve \( -15\div -3\cdot -6 \)

First, divide \( -15 \) by \( -3 \): \( -15\div -3 = 5 \) (negative divided by negative is positive). Then multiply \( 5 \) and \( -6 \): \( 5\times(-6)=-30 \) (positive times negative is negative).

Step7: Answer for "When multiplying 2 integers and they are both negative..."

From the rule \( (-a)\times(-b)=ab \), the answer is positive.

Step8: Answer for "When dividing 2 integers and 1 of them is negative..."

From the rule \( \frac{a}{-b}=-\frac{a}{b} \) or \( \frac{-a}{b}=-\frac{a}{b} \), the answer is negative.

Step9: Answer for "When multiplying with integers, a negative times a positive or a positive times a negative..."

From the rule \( a\times(-b)=-ab \) or \( (-a)\times b=-ab \), the product will be negative.

Step10: Answer for "Multiplication/Division: different signs are..."

From the rules \( a\times(-b)=-ab \), \( (-a)\times b=-ab \), \( \frac{a}{-b}=-\frac{a}{b} \), \( \frac{-a}{b}=-\frac{a}{b} \), different signs give negative.

Step11: Answer for "Multiplication/Division: same signs are..."

From the rules \( (-a)\times(-b)=ab \), \( \frac{-a}{-b}=\frac{a}{b} \), same signs give positive.

Step12: Answer for "When multiplying 3 integers and 2 of them are negative..."

Two negative integers multiply to positive (from \( (-a)\times(-b)=ab \)), then multiplying by the third (positive) integer: positive times positive is positive? Wait, no: Let's take the numbers as \( -a \), \( -b \), \( c \) (where \( a,b,c \) are positive). \( (-a)\times(-b)\times c=(ab)\times c = abc \) (positive). Wait, but wait: 2 negatives and 1 positive. The number of negative factors is 2 (even), so the product is positive? Wait, no, wait: Wait, the rule is that the product of an even number of negative integers is positive, and odd number is negative. So 2 negative integers (even) multiply to positive, then times positive is positive? Wait, no, in the problem, it's 3 integers, 2 negative. So \( (-a)\times(-b)\times c \). First, \( (-a)\times(-b)=ab \) (positive), then \( ab\times c = abc \) (positive). Wait, but let's test with numbers: \( -2\times -3\times 4 \). \( (-2)\times(-3)=6 \), \( 6\times4 = 24 \) (positive). Wait, but if the third number was negative: \( -2\times -3\times -4 \), then \( (-2)\times(-3)=6 \), \( 6\times(-4)=-24 \) (negative, since 3 negative factors, odd number). So for 2 negative factors (even) and 1 positive factor, the product is positive? Wait, no, wait: the number of negative factors is 2 (even), so the product is positive. So the answer is positive? Wait, no, wait the problem says "when multiplying 3 integers and 2 of them are negative". So two negatives and one positive. So the product of two negatives is positive, times positive is positive. So the answer is positive? Wait, but let's check the integer rules: the sign of the product is determined by the number of negative factors. If there are an even number of negative factors, the product is positive; odd number, negative. So 2 negative factors (even) → positive product. So the answer is positive.

Answer:

1. \( -12 \)
2. \( -6 \)
3. \( 72 \)
4. \( 6 \)
5. \( 24 \)
6. \( -30 \)
7. positive
8. negative
9. negative
10. negative
11. positive
12. positive