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1.3 Operations on Real Numbers and Order of Operations

Adding Real Numbers To add two real numbers 1. with the same sign, add their absolute values. Use their common sign as the sign of the answer. 2. with different signs, subtract their absolute values. Give the answer the same sign as the number with the larger absolute value.

Example Add. a. ( ‒ 8) + ( ‒ 3) = ‒ 11 Same sign b. ( ‒ 7) + 1 = ‒ 6 Different signs c.Different signs d. ( ‒ 12.6) + ( ‒ 1.7) = ‒ 14.3Same signs

Subtracting Two Real Numbers If a and b are real numbers, then a – b = a + (– b). Subtracting Real Numbers

Subtract. a. –6 – 5 = –6 + (–5) = –11 b. 7 – (–8) = = 15 c. 4 – 9 = 4 + (–9) = –5 Examples

Subtract. a. 4 ‒ 7 = ‒ 3 b. ‒ 8 ‒ ( ‒ 9) = 1 c. (–5) – 6 – (–3) = ‒ 8 Example

Subtract. a. 6.9 ‒ ( ‒ 1.8) = = 8.7 b. Examples

Multiplying Real Numbers 1. The product of two numbers with the same sign is a positive number. 2. The product of two numbers with different signs is a negative number. Multiplying Real Numbers

Multiply. a. 4(–2) = –8 b. ‒ 7( ‒ 5) = 35 c. 9( ‒ 6.2) = ‒ 55.8 d. Examples

Product Property of 0 a · 0 = 0. Also 0 · a = 0. Example: Multiply. –6 · 0 –6 · 0 = 0 Example: Multiply. 0 · · 125 = 0

Quotient of Two Real Numbers The quotient of two numbers with the same sign is positive. The quotient of two numbers with different signs is negative. Division by 0 is undefined.

Divide. a. b. c. Example

Examples a. Find the quotient. b. Find the quotient.

If a and b are real numbers, and b  0, Simplifying Real Numbers

Exponents Exponents that are natural numbers are shorthand notation for repeating factors. 3 4 = 3 · 3 · 3 · 3 3 is the base 4 is the exponent (also called power) Note by the order of operations that exponents are calculated before other operations.

Evaluate. a. (–2) 4 = (–2)(–2)(–2)(–2) = 16 b. ‒ 7 2 = ‒ (7 ·7) = ‒ 49 Example

Evaluate each of the following expressions = 3 · 3 · 3 · 3 = 81 (–5) 2 = (– 5)(–5)= 25 –6 2 = – (6)(6)= –36 (2 · 4) 3 = (2 · 4)(2 · 4)(2 · 4)= 8 · 8 · 8= · 4 2 = 3 · 4 · 4= 48 Example

Square Roots Opposite of squaring a number is taking the square root of a number. A number b is a square root of a number a if b 2 = a. In order to find a square root of a, you need a number that, when squared, equals a.

The principal (positive) square root is noted as The negative square root is noted as Principal Square Roots

Find the square roots. a. b. c. d. not a real number Example

The Order of Operations Order of Operations Simplify expressions using the order that follows. If grouping symbols such as parentheses are present, simplify expression within those first, starting with the innermost set. If fraction bars are present, simplify the numerator and denominator separately. 1. Evaluate exponential expressions, roots, or absolute values in order from left to right. 2. Multiply or divide in order from left to right. 3. Add or subtract in order from left to right.

Use order of operations to evaluate each expression. a. b. Example

Simplify each expression. a. ‒ 9 – – ( ‒ 7) = ‒ 9 + (–5) = 4 b. Example

Evaluate: Write 3 2 as 9. Divide 9 by 3. Add 3 to 6. Divide 9 by 9.

Evaluating Expressions Example: Evaluate the expression 4 + (4 2 – 13) 4 – 3. Evaluate the exponent inside the parentheses. Work inside the parentheses. Evaluate the exponent. Add. Subtract. = 4 + (3) 4 – 3 = – 3 = 85 – 3 = 82 = 4 + (16 – 13) 4 – (4 2 – 13) 4 – 3

Evaluate each of the following expressions. Example a.) Find 3x 2 when x = 5. b.) Find –2x 2 when x = –1. 3x 2 = 3(5) 2 = 3(5 · 5)= 3 · 25 –2x 2 = –2(–1) 2 = –2(–1)(–1)= –2(1) = 75 = –2

Find the value of the expression when x = 4 and y = ‒ 3. Example

(a) 5x – 2 for x = 8 Evaluate each expression for the given value. (b) 3a 2 + 2a + 4 for a = – 4 5(8) – 2 = 40 – 2 = 38 = 3(– 4) 2 + 2(– 4) + 4 = 3(16) + (– 8) + 4 = 44