Everything I know about exponents

MH9: Exponents Project

1. Represent repeated multiplication with exponents.

EX: a x a x a x a = a⁴    EX: 2 x 2 x 2 x 2 x 2 = 2⁵

2. Describe how powers represent repeated multiplication.

In a power, the base represents the number you multiply by itself. The exponent represents the number of times you multiply the base by itself. So if the repeated multiplication is 6 x 6 x 6, that means the base is 6 and the exponent is 3, since there are three 6’s multiplied together. So 6 x 6 x 6 equals 6^3.

3. Demonstrate the difference between the exponent and the base by building models of a given power, such as 2³ and 3².

As I said before, the base is the number you multiply by itself and the exponent is the number of times you multiply the base. In models, the base is the side length, and the exponent 2 represents the area and the exponent 3 represents the volume.

EX:

3² = 

2³ =  

 

 

 

 

4. Demonstrate the difference between two given powers in which the exponent and the base are interchanged by using repeated multiplication, such as 2³ and 3².

Again, the base is the number you multiply by itself and the exponent is the number of time you multiply the base.

EX: 2³ = 2 x 2 x 2 = 8 (Three 2’s multiplied)           3² = 3 x 3 = 9 (Two 3’s multiplied)

5. Evaluate the powers with integral bases (excluding base 0) and whole number exponents.

EX: 2⁵ = 2 x 2 x 2 x 2 x 2 = 32     EX: 5⁵ = 5 x 5 x 5 x 5 x 5 = 3125       EX: 10⁵ = 10 x 10 x 10 x 10 x 10 = 100 000

   2⁴ = 2 x 2 x 2 x 2 = 16             5⁴ = 5 x 5 x 5 x 5 = 625               10⁴ = 10 x 10 x 10 x 10 = 10 000

   2³ = 2 x 2 x 2 = 8                 5³ = 5 x 5 x 5 = 125                 10³ = 10 x 10 x 10 = 1000

   2² = 2 x 2 = 4                    5² = 5 x 5 = 25                       10² = 10 x 10 = 100

 

EX: (-3)⁵ = (-3)(-3)(-3)(-3)(-3) = -243             EX: -6⁵ = -1 x 6 x 6 x 6 x 6 x 6 = -7776

   -3⁴ = -1 x 3 x 3 x 3 x 3 = -81                    (-6)⁴ = (-6)(-6)(-6)(-6) = 1296

   (-3)³ = (-3)(-3)(-3) = -27                       -6³ = -1 x 6 x 6 x 6 = -216

   -3² = -1 x 3 x 3 = – 9                           (-6)² = (-6)(-6) = 36

   (-3)¹ = (-3)                                   -6¹ = -1 x 6 = -6

6. Explain the role of parentheses in powers by evaluating a given set of powers such as (-2)⁴, (-2⁴) and -2⁴.

When a quantity in parentheses is raised to a power, the exponent applies to everything inside the parentheses.

EX: (-n)² = (-n)(-n)                 EX: (-2)⁴ = (-2)(-2)(-2)(-2) = 16

   (-n²) = (-1 x n x n)                  (-2⁴) = (-1 x 2 x 2 x 2 x 2) = -16

   -n² = -1 x n x n                     -2⁴ = -1 x 2 x 2 x 2 x 2 = -16

   -(-n)² = -1 x (-n) x (-n)               -(-2)⁴ = -1 x (-2) x (-2) x (-2) x (-2) = -16

7. Explain the exponent laws for multiplying and dividing powers with the same base.

When multiplying powers of the same base; keep the base, add the exponents and multiply the coefficients.

EX: ab^x x cb^y = a x c (b^{x +y})

EX: 4(2⁴) x 2(2⁶) = 4 x 2 (2⁴⁺⁶) = 8 (2¹⁰) = 8 x 1024 = 8192

When dividing powers of the same base; keep the base, subtract the exponents and divide the coefficients.

EX:

 

 

 

EX:

 

 

 

8. Explain the exponent laws for raising a product and quotient to an exponent.

When raising a power to an exponent; keep the base, multiply the exponents and the exponent also applies to the coefficients.

EX: (a x b^c)^d = a^d x b^{c x d}      EX: (4 x 2⁴)² = 4² x 2^{4 x 2} = 4² x 2⁸ = 16 x 256 = 4096

9. Explain the law for powers with an exponent of zero.

When a power is raised to a zero exponent the answer is one, except when the base is zero.

EX: a⁰ = 1   a 0

EX: 0⁰ = 0   0⁰ 1

10. Use patterns to show that a power with an exponent of zero is equal to one.

When the exponent decreases by one, the answer is divided by the base. This does not work when the base is zero.

EX: 2⁵ = 32           EX: 3⁵ = 243           EX: 0⁵ = 0

   2⁴ = 16 (2)           3⁴ = 81 (3)               0⁴ = 0

   2³ = 8 (2)             3³ = 27 (3)               0³ = 0

   2² = 4 (2)             3² = 9 (3)               0² = 0

   2¹ = 2 (2)            3¹ = 3 (3)               0¹ = 0

   2⁰ = 1               3⁰ = 1                0⁰ = 0

11. Explain the law for powers with negative exponents.

Any base (except zero) raised to a negative exponent equals the reciprocal of the base raised to a positive exponent.

EX: a^x a^y = a^x-y = a^-y = 1/a^y     EX: a^-x = 1/a^x     EX: (a/b)^-x = (b/a)^x

EX: 2² 2⁴ = 2^2-4 = 2^-2 = 1/2² = 1/4       EX: 6^-3 = 1/6³     EX: (2/3)^-4 = (3/2)⁴ = 12/8

12. Use patterns to explain the negative exponent law.

When the exponent decreases by one, the answer is divided by the base. This does not work when the base is zero.

EX: 2² = 4 (2)       EX: 3² = 9 (3)       EX: 10² = 100 (10)

   2¹ = 2 (2)           3¹ = 3 (3)           10¹ = 10 (10)

   2⁰ = 1 (2)           3⁰ = 1 (3)           10⁰ = 1 (10)

   2^-1 = 1/2 (2)         3^-1 = 1/3 (3)       10^-1 = 1/10 (10)

   2^-2 = 1/4              3^-2 = 1/9            10^-2 = 1/100

13. I can apply the exponent laws to powers both integral and variable bases.

Product Law:                 Product Law with coefficients:        Quotient Law:

b^m x bⁿ = b^m+n                         cb^m x dbⁿ = c x d (b^m+n)                b^m bⁿ = b^m-n

b² x b⁶ = b²⁺⁶ = b⁸                   5b³ x 3b² = 5 x 3 (b³⁺²) = 15(b⁵)          b⁴ b² = b^4-2 = b²

3² x 3⁴ = 3²⁺⁴ = 3⁶                     2(3²) x 3(3³) = 2 x 3 (3²⁺³) = 6(3⁵)         3⁶ 3² = 3^6-2 = 3⁴

(-2)⁴ x (-2)³ = (-2)⁴⁺³ = (-2)⁷       3(-2)² x 2(-2)⁵ = 3 x 2 (-2)²⁺⁵ = 6(-2)⁷           (-4)⁵ (-4)³ = (-4)^5-3 = (-4)²

-4² x -4⁷ = -4²⁺⁷ = -4⁹                 6(-5³) x 2(-5⁴) = 6 x 2 (-5³⁺⁴) = 12(-5⁷)     (-2⁷) (-2⁴) = (-2^7-4) = (-2³)

Quotient Law with coefficients:                      Power Law:                      Zero Exponent Law:

(b^m)ⁿ = b^{m x n}                              b⁰ = 1           

(b³)² = b^{3 x 2} = b⁶                      7⁰ = 1

(4²)³ = 4^{2 x 3} = 4⁶                       (-5)⁰ = 1

((-3)³)² = (-3)^{3 x 2} = (-3)⁶         –  6⁰ = -1

(-5³)⁴ = (-5^{3 x 4}) = (-5¹²)

14. I can identify the error in a simplification of an expression involving powers.

EX:

 

 

 

 

 

 

15. Use the order of operations on expressions with powers.

Use BEDMAS

EX:

 

 

 

 

 

16. Determine the sum and difference of two powers.

EX: a^x + a^y     EX: a^x + b^y     EX: a^x – a^y    EX: a^x – b^y

EX: 4² + 4³ = 16 + 64 = 80       EX: 3⁴ + 1³ = 81 + 1 = 82

EX: 2⁶ – 6² = 64 – 36 = 28       EX: 7² – 7³ = 49 – 343 = -294

17. Identify the error in applying the order of operations in an incorrect solution.

For finding the error in the order of operations involving powers, you need to know the exponent laws and the order of BEDMAS.

EX:

 

 

 

 

 

18. Use powers to solve problems (measurement problems)

EX:

 

 

 

 

 

 

 

 

 

 

 

 

 

19. Use powers to solve problems (growth problems)

EX: A colony of bacteria doubles every hour. There are 12 bacteria now.

1H = 12 x 2 = 24 (x 2)

2H = 12 x 2² = 12 x 4 = 48 (x 2)

3H = 12 x 2³ = 12 x 8 = 96 (x 2)

4H = 12 x 2⁴ = 12 x 16 = 192 (x 2)

5H = 12 x 2⁵ = 12 x 32 = 384

EX: A colony of bacteria triples every hour. There are 21 bacteria now.

1H = 21 x 3 = 63 (x 3)

2H = 21 x 3² = 21 x 9 = 189 (x 3)

3H = 21 x 3³ = 21 x 27 = 567 (x 3)

4H = 21 x 3⁴ = 21 x 81 = 1701 (x 3)

5H = 21 x 3⁵ = 21 x 243 = 5103

20. Applying the order of operations on expressions with powers involving negative exponents and variable bases.

EX:

 

 

 

 

 

 

 

Core Competencies: