4-1 Exponents Course 3 Warm Up Problem of the Day Lesson Presentation

Warm Up Find the product. 1. 5 • 5 • 5 • 5 625 2. 3 • 3 • 3 27 3. (–7) • (–7) • (–7) –343 4. 9 • 9 81

Problem of the Day What two positive integers when multiplied together also equal the sum of the same two numbers? 2 and 2

Learn to evaluate expressions with exponents.

Vocabulary exponential form exponent base power

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. Both 27 and 33 represent the same power. Exponent Base 7 2

Additional Example 1: Writing Exponents Write in exponential form. A. 4 • 4 • 4 • 4 Identify how many times 4 is a factor. 4 • 4 • 4 • 4 = 44 B. (–6) • (–6) • (–6) Identify how many times –6 is a factor. (–6) • (–6) • (–6) = (–6)3 Read –(63) as “-6 to the 3rd power” or “-6 cubed”. Reading Math

Additional Example 1: Writing Exponents Write in exponential form. Identify how many times 5 and d are used as a factor. C. 5 • 5 • d • d • d • d 5 • 5 • d • d • d • d = 52d4

Write in exponential form. Check It Out: Example 1 Write in exponential form. A. x • x • x • x • x Identify how many times x is a factor. x • x • x • x • x = x5 B. d • d • d Identify how many times d is a factor. d • d • d = d3

Write in exponential form. Check It Out: Example 1 Write in exponential form. C. 7 • 7 • b • b Identify how many times 7 and b are used as a factor. 7 • 7 • b • b = 72b2

Additional Example 2: Evaluating Powers Evaluate. A. 35 Find the product of five 3’s. 35 = 3 • 3 • 3 • 3 • 3 = 243 B. (–3)5 Find the product of five –3’s. = (–3) • (–3) • (–3) • (–3) • (–3) (–3)5 = –243 Helpful Hint Always use parentheses to raise a negative number to a power.

Additional Example 2: Evaluating Powers Evaluate. C. (–4)4 Find the product of four –4’s. = (–4) • (–4) • (–4) • (–4) (–4)4 = 256 D. 28 Find the product of eight 2’s. 28 = 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 = 256

Evaluate. A. 74 Find the product of four 7’s. 74 = 7 • 7 • 7 • 7 Check It Out: Example 2 Evaluate. A. 74 Find the product of four 7’s. 74 = 7 • 7 • 7 • 7 = 2401 B. (–9)3 Find the product of three –9’s. = (–9) • (–9) • (–9) (–9)3 = –729

Evaluate. C. –(5)2 = –(5) • (5) –(5)2 = –25 D. 97 Check It Out: Example 2 Evaluate. C. –(5)2 Find the product of two 5’s and then make the answer negative. = –(5) • (5) –(5)2 = –25 D. 97 Find the product of seven 9’s. 97 = 9 • 9 • 9 • 9 • 9 • 9 • 9 = 4,782,969

Additional Example 3: Using the Order of Operations Evaluate x(yx – zy) + x for x = 4, y = 2, and z = 3. y x(yx – zy) + x y Substitute 4 for x, 2 for y, and 3 for z. = 4(24 – 32) + 42 = 4(16 – 9) + 16 Evaluate the exponent. = 4(7) + 16 Subtract inside the parentheses. = 28 + 16 Multiply from left to right. = 44 Add.

Evaluate z – 7(2x – xy) for x = 5, y = 2, and z = 60. Check It Out: Example 3 Evaluate z – 7(2x – xy) for x = 5, y = 2, and z = 60. z – 7(2x – xy) Substitute 5 for x, 2 for y, and 60 for z. = 60 – 7(25 – 52) = 60 – 7(32 – 25) Evaluate the exponent. = 60 – 7(7) Subtract inside the parentheses. = 60 – 49 Multiply from left to right. = 11 Subtract.

Additional Example 4: Geometry Application 1 2 Use the formula (n2 – 3n) to find the number of diagonals in a 7-sided figure. (n2 – 3n) 1 2 (72 – 3 • 7) 1 2 Substitute the number of sides for n. (49 – 3 • 7) 1 2 Evaluate the exponent. (49 – 21) 1 2 Multiply inside the parentheses. (28) 1 2 Subtract inside the parentheses. 14 diagonals Multiply

Additional Example 4 Continued A 7-sided figure has 14 diagonals. You can verify your answer by sketching the diagonals.

(n2 – 3n) (42 – 3 • 4) (16 – 3 • 4) (16 – 12) (4) 2 diagonals Check It Out: Example 4 1 2 Use the formula (n2 – 3n) to find the number of diagonals in a 4-sided figure. (n2 – 3n) 1 2 (42 – 3 • 4) 1 2 Substitute the number of sides for n. (16 – 3 • 4) 1 2 Evaluate the exponents. (16 – 12) 1 2 Multiply inside the parentheses. (4) 1 2 Subtract inside the parentheses. 2 diagonals Multiply.

Check It Out: Example 4 Continued A 4-sided figure has 2 diagonals. You can verify your answer by sketching the diagonals.

Write in exponential form. Lesson Quiz: Part I Write in exponential form. 1. n • n • n • n 4 n 2. (–8) • (–8) • (–8) • (h) (–8)3h 3. Evaluate (–4)4 256 4. Evaluate x • z – yx for x = 5, y = 3, and z = 6. –213

Lesson Quiz: Part II 5. A population of bacteria doubles in size every minute. The number of bacteria after 5 minutes is 15  25. How many are there after 5 minutes? 480