: Algebra I Exponents and Powers
: Goal 1: Evaluate Expressions Containing Exponents An expression like 4^6 is called a power. The exponent 6 represents the number of times the base 4 is used as a factor. 4 = Base 6 = Exponent4^6 = Power Meaning = (4)(4)(4)(4)(4)(4) or 6 factors of 4
: Reading and Writing Powers Express the meaning of the power in words and then with numbers or variables. a. 10^1 ten to the first power or 10b. 4^2 four to the second power, or four squared or (4)(4)c. 5^3 five to the third power, or five cubed or (5)(5)(5) For a number raised to the first power, you usually do not write the exponent 1. For instance, you write 5^1 simply as 5.
: Evaluating Powers Evaluate the expression x^3 when x = 5. x^3 = 5^3 Substitute 5 for x. = (5)(5)(5) Write factors. = 125 Multiply. The value of the expression is 125.
: Grouping Symbols For problems that have more than one operation, it is important to know which operation to do first. Grouping symbols, such as parentheses () or brackets [], indicate the order in which the operations should be performed. Operations within the innermost set of grouping symbols are done first. For instance, the value of the expression (3 * 4) + 7 is not the same as the value of the expression 3(4 + 7). Multiply. Then add. Add. Then multiply. (3 * 4) + 7 = 12 + 7 = 19 3(4 + 7) = 3(11) = 33
: Evaluating an Exponential Expression Evaluate the expression when a = 1 and b = 2. a. (a + b)^2 = (1 + 2)^2 Substitute. = (3)^2 Add within parentheses. = (3)(3) Write factors. = 9 Multiply. b. (a^2) + (b^2) = (1^2) + (2^2) Substitute. = 1 + 4 Evaluate power. = 5 Add.
: Title An exponent applies only to the number, variable, or expression immediately to its left. In the expression 2x^3, the base is x, not 2x. In the expression (2x)^3, the base is 2x, as indicated by the parentheses.
: Exponents and Grouping Symbols Evaluate the expression when x = 4. a. 2x^3 = 2(4^3) Substitute 4 for x. = 2(64) Evaluate power. = 128 Multiply. b. (2x)^3 = (2 * 4)^3 Substitute 4 for x. = 8^3 Multiply within parentheses. = 512 Evaluate power.
: Goal 2: Real-Life Applications of Exponents Exponents are often used in the formulas for area and volume. In fact, the words squared and cubed come from the formula for the area of a square, A = s^2, and the formula for the volume of a cube, V = s^3. Units of area, such as square feet, ft^2, can be written using a second power. Units of volume, such as cubic centimeters, cm^3, can be written using a third power.
: Finding Volume The aquarium has the shape of a cube. Each edge is 2.5 feet long. a. Find the volume in cubic feet.b. How many gallons of water will the cubic aquarium hold? Convert to liquid volume, where one cubic foot holds 7.48 gallons.
: Finding Volume Con. a. V = s^3 Write formula for volume. = 2.5^3 Substitute 2.5 for s. = 15.625 Evaluate power. The volume of the aquarium is 15.625 ft^3. b. V = 15.625 ft^3 (7.48 gal/1 ft^3) Write conversion factor. = 116.875 gal Multiply. A 15.625 cubic foot aquarium will hold 116.875 gallons of water.