Evaluating Algebraic Expressions 4-1Exponents AF2.1 Interpret positive whole-number powers as repeated multiplication and negative whole-number powers as repeated division or multiplication by the multiplicative inverse. Simplify and evaluate expressions that include exponents. Also covered: AF1.2 California Standards

Evaluating Algebraic Expressions 4-1Exponents If a number is in exponential form, the exponent represents how many times the base is to be used as a factor. A number produced by raising a base to an exponent is called a power. 27 and 3 3 are equivalent. 7 Exponent Base 2

Evaluating Algebraic Expressions 4-1Exponents Identify how many times 4 is a factor = 4 4 Write in exponential form. Additional Example 1: Writing Exponents A Read (–6) 3 as “–6 to the 3rd power" or "–6 cubed”. Reading Math Identify how many times –6 is a factor. (–6) (–6) (–6) = (–6) 3 B. (–6) (–6) (–6)

Evaluating Algebraic Expressions 4-1Exponents Identify how many times 5 and d are each used as a factor. Additional Example 1: Writing Exponents C. 5 5 d d d d Write in exponential form. 5 5 d d d d = 5 2 d 4

Evaluating Algebraic Expressions 4-1Exponents Identify how many times x is a factor. x x x x x = x 5 Write in exponential form. Check It Out! Example 1 A. x x x x x Identify how many times d is a factor. d d d = d 3 B. d d d

Evaluating Algebraic Expressions 4-1Exponents Identify how many times 7 and b are each used as a factor. 7 7 b b = 7 2 b 2 Check It Out! Example 1 C. 7 7 b b Write in exponential form.

Evaluating Algebraic Expressions 4-1Exponents A. 3 5 = = Find the product. B. Simplify. Additional Example 2: Simplifying Powers = 1 27

Evaluating Algebraic Expressions 4-1Exponents D. – 2 8 = 256 –2 8 = –( ) = –256 = (–4) (–4) (–4) (–4) (–4) 4 C. (–4) 4 Simplify. Additional Example 2: Simplifying Powers Find the product. Find the product. Then make the answer negative.

Evaluating Algebraic Expressions 4-1Exponents The expression (–4) 4 is not the same as the expression –4 4. Think of –4 4 as –1 ● 4 4. By the order of operations, you must evaluate the exponent before multiplying by –1. Caution!

Evaluating Algebraic Expressions 4-1Exponents A. 7 4 = = Find the product. Simplify. Check It Out! Example 2 Find the product. B. = 1818

Evaluating Algebraic Expressions 4-1Exponents D. – 9 4 = 25 –9 4 = –( ) = –6,561 = (–5) ( – 5) (–5) 2 C. (–5) 2 Evaluate. Find the product. Find the product. Then make the answer negative. Check It Out! Example 2

Evaluating Algebraic Expressions 4-1Exponents Additional Example 3: Using the Order of Operations 4(7) + 16 Substitute 4 for x, 2 for y, and 3 for z. Simplify the powers. Subtract inside the parentheses. Multiply from left to right. 4(2 4 – 3 2 ) (16 – 9) Evaluate x(y x – z y ) + x for x = 4, y = 2, and z = 3. y x(y x – z y ) + x y Add. 44

Evaluating Algebraic Expressions 4-1Exponents Check It Out! Example 3 60 – 7(7) Substitute 5 for x, 2 for y, and 60 for z. Simplify the powers. Subtract inside the parentheses. Multiply from left to right. 60 – 7(2 5 – 5 2 ) 60 – 7(32 – 25) 60 – 49 Evaluate z – 7(2 x – x y ) for x = 5, y = 2, and z = 60. z – 7(2 x – x y ) Subtract. 11

Evaluating Algebraic Expressions 4-1Exponents (7 2 – 3 7) 1212 Additional Example 4: Geometry Application Simplify inside the parentheses. Multiply. Substitute the number of sides for n. Subtract inside the parentheses. 14 diagonals (49 – 21) 1212 (n 2 – 3n) 1212 (28) 1212 Use the expression (n 2 – 3n) to find the number of diagonals in a 7-sided figure. 1212

Evaluating Algebraic Expressions 4-1Exponents (4 2 – 3 4) 1212 Check It Out! Example 4 Simplify inside the parentheses. Multiply Substitute the number of sides for n. Subtract inside the parentheses. 2 diagonals (16 – 12) 1212 (n 2 – 3n) 1212 (4) 1212 Use the expression (n 2 – 3n) to find the number of diagonals in a 4-sided figure. 1212