Digital Lesson Numerical Expressions These are examples of

Digital Lesson Numerical Expressions

These are examples of numerical expressions. grouping symbols A numerical expression is an expression formed from numbers by adding, subtracting, multiplying, dividing, taking powers, taking roots, and using grouping symbols: ( ), [ ], | |, { }, and the horizontal bar as in fractions and radicals. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

To evaluate a numerical expression, carry out the indicated operations to arrive at a single numerical value. Examples: Evaluate. 1. 8 – 5(2) + 1 = 8 – 10 + 1 = – 2 + 1 Value = – 1 2. – 2 + 3(7) = – 2 + 21 Value = 19 3. – 1+5 =7– 1+5 =6+5 = 11 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Value 3

To evaluate a variable expression at a given value, substitute the variable with its value, then evaluate the numerical expression. Examples: 1. Evaluate 2 x + 3 when x = 5. 2(5) + 3 = 13 Value 2. Evaluate 3 x 2 + x – 4 when x = 2. 3(2)2 + (2) – 4 = 3 • 4 + (2) – 4 = 10 3. Evaluate Value when x = 3 and y = 1. (3) (1) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 1 1 Value 4

Adding and Subtracting Signed Numbers Examples: Add or subtract. The signs are the same, so add the absolute values and attach the common sign. 1. – 1 + (– 3) = – 4 -4 -3 -2 -1 0 1 2 3 4 2. – 4 + 6 = 2 -4 -3 -2 -1 0 1 2 3 4 3. – 2 – (– 3) = – 2 + 3 = 1 -4 -3 -2 -1 0 1 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 The signs are different, so subtract the absolute values and attach the sign of the number with the larger absolute value. To subtract, write the expression as the addition of the opposite number. 4 5

Multiplying and Dividing Signed Numbers Examples: Multiply or divide. 1. (– 6) · (– 2) = 12 The signs are the same, so the product is positive. 2. (– 5) · 4 = – 20 The signs are different, so the product is negative. Every division can be written as a multiplication problem. 3. The signs are different, so the quotient is negative. 4. =3 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The signs are the same, so the quotient is positive. 6

The arithmetic of signed numbers and fractions can be used to evaluate numerical expressions. Examples: 1. Evaluate – x + 2 x when x = – 5. – (– 5) + 2(– 5) = – 1 · (– 5) + 2 · (– 5) = 5 + (– 10) Signs are the same, so the product will be positive. Signs are different, so the product will be negative. Signs are different. Subtract. = – 5 2. Evaluate when x = – 2. (– 2) Division by zero is undefined. This expression is undefined when x = – 2. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7

Adding and Subtracting Fractions Examples: Add or subtract. 1. 2. 3. The denominators are the same, so add the numerators. Place the sum over the common denominator. The denominators are the same, so subtract the numerators. Find a common denominator. The common denominator is 8. Subtract the numerators and place the result over the denominator. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8

Multiplying and Dividing Fractions Example: Multiply or divide. 1. Multiply the numerators. Multiply the denominators. Place the product of the numerators over the product of the denominators. 2. The sign rules for fractions are the same as those for integers. Write as a multiplication by the reciprocal of the divisor. Multiply the numerators. Multiply the denominators. The result is negative since the signs are different. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9

The Order of Operations: 1. Perform all operations within grouping symbols as they occur from the inside out. Grouping symbols can be parentheses ( ), brackets [ ], the absolute value symbol | |, and the horizontal bar used in fractions and radicals. 2. Simplify all exponential expressions. 3. Do all multiplications and divisions as they occur from left to right. 4. Do all additions and subtractions as they occur from left to right. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10

Examples: Simplify. 1. 12 – 6 + 2 – 1 =8– 1 Subtract. Add. =7 Subtract. 2. 3 · 5 – 2 = 15 – 2 Multiply. = 13 3. 3 · (5 – 2) = 3 · 3 The grouping symbols change the order of operations, changing the value of the expression. =9 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Subtract. Do subtraction within the grouping symbols. Multiply. 11

Examples: Simplify. 1. 20 Multiply. 3 Divide. Add. 2. 8 (2) = 18 – 4 = 14 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Evaluate expressions inside grouping symbols from the inside out. Simplify the exponential expression. Subtract. 12

Examples: Simplify. 54 3 1. Evaluate above and below the fraction bar. 9 Simplify the exponential expression. 18 Do left-most division. 2 Divide. Subtract. 2. = 5 – [(1 + 3)]2 Square root = 5 – (4)2 Do addition within grouping symbols. = 5 – 16 = 11 Simplify exponent. Subtract. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 13