Powers and Exponents

Vocabulary and Power Notation

  • A power is defined as a product of repeated factors.

  • The base is the repeated factor in a power.

  • The exponent indicates how many times the base is used as a factor.

  • Terminology for reading powers:

    • 323^2: Three squared, or three to the second
    • 333^3: Three cubed, or three to the third
    • 343^4: Three to the fourth
    • 353^5: Three to the fifth

  • Writing expressions as powers involves counting the occurrences of the factor:

    • 7⋅7⋅7⋅7⋅7=757 \cdot 7 \cdot 7 \cdot 7 \cdot 7 = 7^5
    • 12×12×12=12312 \times 12 \times 12 = 12^3
    • 100×100×100×100×100×100=1006100 \times 100 \times 100 \times 100 \times 100 \times 100 = 100^6

Evaluating Powers and Perfect Squares

  • To find the value of a power, use repeated multiplication:

    • 72=7⋅7=497^2 = 7 \cdot 7 = 49
    • 53=5⋅5⋅5=1255^3 = 5 \cdot 5 \cdot 5 = 125

  • A perfect square is the square of a whole number.

  • Identifying perfect squares:

    • 6464 is a perfect square because 82=648^2 = 64.
    • 2020 is not a perfect square as no whole number squared equals 2020.
    • 11 is a perfect square because 12=11^2 = 1.

Area Applications and Data Modeling

  • The area of a square is equal to its side length squared: Area=(side length)2\text{Area} = (\text{side length})^2.

  • In a data display of 100100 people's movie preferences (Horror, Science fiction, Comedy, Drama, Fantasy, Adventure, Action), specific sections can be calculated using square areas:

    • The "Action" section is a square with a side length of 66 units.
    • Area=62=36\text{Area} = 6^2 = 36. Therefore, 3636 people prefer action movies.

Practice and Discussion

  • Write the product as a power:

    • 1. 15×15×15×1515 \times 15 \times 15 \times 15
    • 2. 6⋅6⋅6⋅6⋅6⋅66 \cdot 6 \cdot 6 \cdot 6 \cdot 6 \cdot 6

  • 3. SMP.7: Write 13⋅13⋅7⋅13⋅713 \cdot 13 \cdot 7 \cdot 13 \cdot 7 as a product of two powers, if possible. Explain your reasoning.

  • Find the value of the power:

    • 4. 636^3
    • 5. 929^2

  • Determine whether the number is a perfect square:

    • 6. 2525
    • 7. 9999

  • 8. SMP.3: Your friend says that 2020 is not a perfect square because 42=164^2 = 16 and 52=255^2 = 25. Complete your friend's reasoning.

  • 9. In Example 4, how many people said they prefer comedies?

    1. SMP.1: The die is in the shape of a cube. The total area of its faces is 1,536 square millimeters1,536\,\text{square millimeters}. What is the side length of a face of the die? Explain.