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Understanding Exponents: Evaluation, Powers, and Expressions

This lesson focuses on the concept of exponents, teaching students how to evaluate expressions with exponents and write numbers in exponential form. Students will learn about bases and exponents, simplifying expressions, and solving problems related to powers of positive and negative integers. Examples include finding products of repeated factors, explaining terms such as 'power', and applying methods to evaluate and simplify expressions that contain exponents. Engaging with these concepts will enhance students' mathematical skills and understanding of exponential growth.

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Understanding Exponents: Evaluation, Powers, and Expressions

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  1. 2-6 Exponents Course 3 Warm Up Problem of the Day Lesson Presentation

  2. 2-6 Exponents Course 3 Warm Up Find the product. 1. 5 • 5 • 5 • 5 625 27 2. 3 • 3 • 3 –343 3. (–7) • (–7) • (–7) 4. 9 • 9 81

  3. 2-6 Exponents Course 3 Problem of the Day What two positive integers when multiplied together also equal the sum of the same two numbers? 2 and 2

  4. 2-6 Exponents Course 3 Learn to evaluate expressions with exponents.

  5. 2-6 Exponents Course 3 Vocabulary power exponential form exponent base

  6. 2-6 Exponents Course 3 The term 27 is called a power. If a number is in exponential form, the exponent represents how many times the base is to be used as a factor. Exponent Base 7 2

  7. 2-6 Exponents 4 • 4 • 4 • 4 = 44 d•d•d•d•d = d5 Reading Math Read 44 as “4 to the 4th power.” Course 3 Additional Example 1A & 1B: Writing Exponents Write in exponential form. A. 4 • 4 • 4 • 4 Identify how many times 4 is a factor. B. d • d • d • d • d Identify how many times d is a factor.

  8. 2-6 Exponents Course 3 Additional Example 1C & 1D: Writing Exponents Write in exponential form. C. (–6) • (–6) • (–6) Identify how many times –6 is a factor. (–6) • (–6) • (–6) = (–6)3 D. 5 • 5 5 • 5 = 52 Identify how many times 5 is a factor.

  9. 2-6 Exponents x • x • x • x • x= x5 d•d•d = d3 Course 3 Try This: Example 1A & 1B Write in exponential form. A. x • x • x • x • x Identify how many times x is a factor. B. d • d • d Identify how many times d is a factor.

  10. 2-6 Exponents Course 3 Try This: Example 1C & 1D Write in exponential form. C. (–3) • (–3) • (–3) • (–3) Identify how many times –3 is a factor. (–3) • (–3) • (–3) • (–3) = (–3)4 D. 7 • 7 Identify how many times 7 is a factor. 7 • 7 = 72

  11. 2-6 Exponents A. 35 B. (–3)5 = (–3) • (–3) • (–3) • (–3) • (–3) (–3)5 Course 3 Additional Example 2A & 2B: Evaluating Powers Evaluate. Find the product of five 3’s. 35 = 3 • 3 • 3 • 3 • 3 = 243 Find the product of five –3’s. = –243 Helpful Hint Always use parentheses to raise a negative number to a power.

  12. 2-6 Exponents D. 28 = (–4) • (–4) • (–4) • (–4) (–4)4 C. (–4)4 Course 3 Additional Example 2C & 2D: Evaluating Powers Continued Evaluate. Find the product of four –4’s. = 256 Find the product of eight 2’s. 28= 2 • 2 • 2 • 2 • 2 • 2 • 2 • 2 = 256

  13. 2-6 Exponents A. 74 B. (–9)3 = (–9) • (–9) • (–9) (–9)3 Course 3 Try This: Example 2A & 2B Evaluate. Find the product of four 7’s. 74 = 7 • 7 • 7 • 7 = 2401 Find the product of three –9’s. = –729

  14. 2-6 Exponents D. 97 = (–5) • (–5) (–5)2 C. (–5)2 Course 3 Try This: Example 2C & 2D Evaluate. Find the product of two –5’s. = 25 Find the product of seven 9’s. 97 = 9 • 9 • 9 • 9 • 9 • 9 • 9 = 4,782,969

  15. 2-6 Exponents Course 3 Additional Example 3: Simplifying Expressions Containing Powers Simplify (25 – 32) + 6(4). = (32 – 9) + 6(4) Evaluate the exponents. = (23) + 6(4) Subtract inside the parentheses. Multiply from left to right. = 23 + 24 Add from left to right. = 47

  16. 2-6 Exponents Course 3 Try This: Example 3 Simplify (32 – 82) + 2 • 3. = (9 – 64) + 2 • 3 Evaluate the exponents. = (–55) + 2 • 3 Subtract inside the parentheses. Multiply from left to right. = –55 + 6 = –49 Add from left to right.

  17. 2-6 Exponents 1 2 1 2 1 2 1 2 1 2 1 2 (n2 – 3n) (72 – 3 • 7) (49 – 3 • 7) (49 – 21) (28) Course 3 Additional Example 4: Geometry Application Use the formula (n2 – 3n) to find the number of diagonals in a 7-sided figure. Substitute the number of sides for n. Evaluate the exponent. Multiply inside the parentheses. Subtract inside the parentheses. 14 diagonals Multiply

  18. 2-6 Exponents Course 3 Additional Example 4 Continued Verify your answer by sketching the diagonals. 14 Diagonals

  19. 2-6 Exponents 1 2 1 2 1 2 1 2 1 2 1 2 (n2 – 3n) (42 – 3 • 4) (16 – 3 • 4) (16 – 12) (4) Course 3 Try This: Example 4 Use the formula (n2 – 3n) to find the number of diagonals in a 4-sided figure. Substitute the number of sides for n. Evaluate the exponents. Multiply inside the parentheses. Subtract inside the parentheses. 2 diagonals Multiply

  20. 2-6 Exponents Course 3 Try This: Example 4 Continued Verify your answer by sketching the diagonals. 2 diagonals

  21. 2-6 Exponents n 4 Course 3 Lesson Quiz: Part 1 Write in exponential form. 1. n•n•n•n 2. (–8) • (–8) • (–8) (–8)3 3. Evaluate (–4)4 256 4. Simplify 99 – 3(4 • 23). 3

  22. 2-6 Exponents Course 3 Lesson Quiz: Part 2 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

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