1.6 Sub Chapter Notes

 

Operations Considering Signs • Adding with like signs involves adding the values and using their common sign. • Adding with unlike signs involves subtracting the values and using the sign of the larger value. • Multiplying with two negative signs produces a positive answer. Adding when the signs of the numbers are the same is just a matter of combining the values and keeping their common sign. In this kind of problem, let the number line be your friend. This example shows adding two numbers that both have positive signs.

This example shows adding two numbers that both have negative signs.

When adding numbers whose signs do not match, the values actually diminish each other. Subtract the two values and give the answer the sign of the larger value. It never hurts to do a quick sketch with a number line if there is any question about the result.

Multiplying with two negatives always produces a positive answer. Notice in this example, the negative in front of the parentheses multiplies with the negative inside the parentheses, and the number becomes positive.

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Prime and Composite Numbers and Their Roots • A prime number has exactly two distinct factors, itself and one. • A composite number is composed of prime numbers. • Taking the root is like undoing the exponent. A prime number is one that cannot be broken into smaller whole numbers multiplying together. For example, the number 7 is the answer only when 1 and 7 multiply together. The number 11 is the answer only when 1 and 11 multiply together. Prime numbers are useful because all natural numbers greater than one either are prime or they are products of primes. A composite number is the product of prime numbers. This example shows the number that is the product of three 2s and one 3. 24 is the only real number existing that is the product of three 2s and one 3. The root of a given number is the value that multiplies with itself the indicated number of times to produce the given number. A square root indicates that two of some number multiplied with themselves to produce the given number. In this example, 36 is the given number. The question asks what number multiplies with itself to produce 36. Knowing that two 6s multiply to produce 36 reveals that the square root of 36 is 6. The cube root of a number multiplies with itself and then multiplies again to produce the given number. In this example, 125 is the given number and you are looking for the cube root. Knowing that 125 is the product when three 5s multiply together reveals that the cube root of 125 is 5, i.e. 5 · 5 · 5 = 125. Remember: Exponents tell how many times a number will be multiplied. Roots tell how many times a number has been multiplied.

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Order of Operations • Find distance by subtracting on a number line. • Follow the order of operations to simplify expressions. One way to think about the distance separating two values is to consider the number of spaces separating the values on a number line. To find the distance between the two values, subtract one from the other. Remember, you are counting spaces and your distance is always positive.

In simplifying any expression, do the work in a specific order. Following this rule is your road to accurate answers.

To simplify an expression work your way through each step in the order of operations: Step 0: There are no expressions in parentheses for this problem, so move on. Step 1: This is a fraction so simplify the numerator and the denominator separately from each other. Step 2: In the numerator, take the square root of 49 and square 6. In the denominator, take the square root of 4. Step 3: In the numerator, multiply –3 with 7. In the denominator, multiply –7 with 2. Step 4: In the numerator, combine –21 with –36. In the denominator, combine 5 with –14. Follow this sequence for simplifying any expression. Then, check for any other possible simplification. In this case –3 can be factored out to reduce the fraction.

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