Exponent Rules

Bell Work
3/18/2024

Learning Targets

•I can simplify expressions involving exponents.

•I can define and use zero and negative exponents

•I can apply rules of multiplying powers.

•I can apply rules of dividing powers.

Look for Patterns:

Bell Work
3/18/2024

Use order of operations to simplify each expression:

1.

9 ÷ 3 + 42

2. −3 4 + 6 ÷ 22

Evaluate each expression for 𝑎 = −2, 𝑎𝑛𝑑 𝑏 = 5

1.

𝑎𝑏2

2. 𝑎3+ 𝑏3

3. 𝑏 − 3𝑎2

Learning Targets

• I can simplify expressions involving exponents.

• I can define and use zero and negative exponents

• I can apply rules of multiplying powers.

• I can apply rules of dividing powers.

Look for Patterns:

𝒙

𝒚 = 𝟐𝒙

3

23=

2
1
0
−1
−2

1.What pattern do you see as you complete

2𝑥column?

2.What do you notice in the row where

“zero” is the exponent?

3.How are the terms 2𝑥𝑎𝑛𝑑 2−𝑥related?

Zero and Negative Exponents

1. 40=

2.

−50=

3. −50=

1. 4−2=

2.

−5−2=

3.

1

5−2 =

𝟏

𝟏

−𝟏

𝟏
𝟒𝟐

𝟏

(−𝟓)𝟐

𝟓𝟐

Multiplying Powers with same base

For every number, 𝑎 ≠ 0,

𝒂𝒎∗ 𝒂𝒏= 𝒂𝒎+𝒏

Examples: What is each expression written using each base only once
(simplified form)?

1. 124∗ 123=

2. −53−57=

3. 0.1−20.1−30.14=

4. 4𝑧5∗ 9𝑧−12=

5. 2𝑎 ∗ 9𝑏4∗ 3𝑎2=

6. 𝑗2∗ 𝑘−2∗ 12𝑗 =

𝟏𝟐𝟒+𝟑= 𝟏𝟐𝟕

(−𝟓)𝟑+𝟕= (−𝟓)𝟏𝟎

(𝟎. 𝟏)−𝟐−𝟑+𝟒= (𝟎. 𝟏)−𝟏= 𝟏

𝟎. 𝟏

𝟑𝟔(𝐳)𝟓−𝟏𝟐= 𝟑𝟔(𝐳)−𝟕=𝟑𝟔

𝒛𝟕

𝟓𝟒(𝒂)𝟏+𝟐𝒃𝟒= 𝟓𝟒𝒂𝟑𝒃𝟒

𝟏𝟐(𝒋)𝟐+𝟏𝒌−𝟐= 𝟏𝟐𝒋𝟑𝒌−𝟐=𝟏𝟐𝒋𝟑

𝒌𝟐

You Try: What is each expression written
using each base only once (simplified form)?

1. 83∗ 86=

2.

0.5−30.5−7=

3. 9−3∗ 92∗ 96=

4. 5𝑥4∗ 𝑥9=

5. −4𝑐3∗ 7𝑑2∗ 2𝑐−2=

6. 4𝑐 ∗ 3𝑑5∗ 2𝑐3=

Bell Work
3/21/2024

Simplify (your answer should contain only positive exponents)

1.

4−1∗ 43

2.

3−4∗ 3−2

3.

2𝑥3𝑦−1∗ 2𝑥−3𝑦4

4.

4𝑥3𝑦−1∗ 4𝑦−2

5.

3𝑣𝑢2∗ 4𝑢−1𝑣2

6.

4𝑥3𝑦4𝑧3∗ 4𝑥−2𝑦4

7.

4𝑝−1𝑞−4𝑟3∗ 4𝑝−3𝑞3𝑟3

Adding Fractions

•

1
2+

3
2=

•

1
4+

3
2=

•

2
3+

3
2=

•

2
3+ 3 =

Same Denominator
•

Keep the base

•

Add the numerators

Not Same Denominator

•

Make the denominators same.

•

Add the numerators

𝟏 + 𝟑

𝟐
= 𝟒

𝟐 = 𝟐

𝟏
𝟒 + 𝟔

𝟒 = 𝟕

𝟒

𝟒
𝟔+ 𝟗

𝟔= 𝟏𝟑

𝟔

𝟐
𝟑+ 𝟑

𝟏=

𝟐
𝟑 + 𝟗

𝟑 =𝟏𝟏

𝟑

Integer Exponents and Rational Exponents

• Integer Exponents:

𝑥3

𝑥10

• Rational Exponents:

𝑥

1
2 =

𝑥

𝑥

1
3 =3𝑥

𝑥

3
2 = 𝑥

1
2 ∗ 𝑥

1
2 ∗ 𝑥

3
2

𝑥

2
3 = 𝑥

1
3 ∗ 𝑥

1
3

Simplifying and expression with Powers

𝑥

2
3

3
5𝑥−3

5 = 𝑥

𝟐
𝟑∗𝟑

𝟓𝑥−3

5

= 𝑥

𝟔
𝟏𝟓𝑥−3

5

= 𝑥

𝟐
𝟓𝑥−3

5

= 𝑥

𝟐
𝟓−3

5

= 𝑥−𝟏

𝟓
= 𝟏

𝒙

𝟏
𝟓

Raising a Product to a Power

4𝑚

1
2

3

= 4𝑚

1
2 ∗ 4𝑚

1
2 ∗ 4𝑚

1
2 = 43𝑚

3
2 = 64𝑚

3
2

 Words: To raise a product to a power, raise each factor to the power

and multiply.

 Algebra: 𝒂𝒃𝒏= 𝒂𝒏𝒃𝒏where 𝑎 ≠ 0, 𝑏 ≠ 0 and n is a rational

number.

Examples

3𝑥4= 34∗ 𝑥4= 81𝑥4

4𝑏

3
2 = 4

3
2𝑏

3
2 = 8𝑏

3
2

What is the simplified form of:

𝑛

1
2

10

4𝑚𝑛−2

3

3

𝑏

1
3

6

2𝑎𝑏−1

2

4

= 𝑛

𝟏
𝟐∗𝟏𝟎43𝒎𝟑𝒏−𝟐

𝟑∗𝟑

=𝑛𝟓43𝒎𝟑𝒏−𝟐

=64𝑛𝟑𝒎𝟑

=𝑏𝟐24𝒂𝟒𝒃−𝟐

= 𝑏

𝟏
𝟑∗624𝒂𝟒𝒃−𝟏

𝟐∗𝟒

=16𝑏𝟐𝒂𝟒𝒃−𝟐

=16𝑏𝟎𝒂𝟒

=16𝒂𝟒

You Try:

1.

𝑥−2 23𝑥𝑦5 4

2.

3 𝑐

5
2

4

𝑐2 3

3.

6𝑎𝑏35𝑎−3 2

Division Properties of Exponents

45

43 = 4 ∗ 4 ∗ 4 ∗ 4 ∗ 4

4 ∗ 4 ∗ 4
= 42

• Words: To divide powers with the same base, subtract the exponents
• Algebra:

𝑎𝑚

𝑎𝑛 = 𝑎𝑚−𝑛

𝑤ℎ𝑒𝑟𝑒 𝑎 ≠ 0 and m and n are rational numbers.

Examples:

26

22 = 2𝟔−𝟐 = 𝟐𝟒

𝑥4

𝑥7 = 𝑥𝟒−𝟕 = 𝑥−3= 𝟏

𝒙𝟑

𝑠

3
4

𝑠

1
2

= 𝑠

𝟑
𝟒−𝟏

𝟐 = 𝑠

3
4−2

4

= 𝑠

1
4

Dividing Algebraic Expressions

𝑥

5
2

𝑥2

𝑐

12
5

𝑐2
= 𝑥

𝟓
𝟐−𝟐

= 𝑥

𝟓
𝟐−𝟐

𝟏

= 𝑥

𝟓
𝟐−𝟒

𝟐

= 𝒙

𝟏
𝟐

= 𝑐

𝟏𝟐
𝟓 −𝟐

𝟏

= 𝑐

𝟏𝟐
𝟓 −𝟏𝟎

𝟓

= 𝒄

𝟐
𝟓

Raising a Quotient to a Power

𝑥
𝑦

3

=𝑥

𝑦 ∗ 𝑥

𝑦 ∗ 𝑥

𝑦 = 𝑥3

𝑦3

• Words: To raise a quotient to a power, raise the numerator and

denominator to the power and simplify.

• Algebra:

𝑎
𝑏

𝑛

=

𝑎𝑛

𝑏𝑛 , 𝑤ℎ𝑒𝑟𝑒 𝑎 ≠ 0, 𝑏 ≠ 0 and n is a rational number

𝑎
𝑏

−𝑛

=
1
𝑎
𝑏

𝑛= 1 ∗ 𝑏𝑛

𝑎𝑛 =
𝑏
𝑎

𝑛

Simplify:

2𝑥6

𝑦4

−3

=
𝑦4

2𝑥6

𝟑

= 𝑦𝟏𝟐

2𝟑𝑥𝟏𝟖

= 𝑦𝟏𝟐

8𝑥𝟏𝟖

3𝑐3

𝑑2

−4

=
𝑑2

3𝑐3

4

=
𝑑8

34𝑐12

=
𝑑8

81𝑐12

Simplify

•

𝑥

1
3

3

3

•

𝑚
𝑛

−3

•

𝑎

3
4

𝑎5

4

•

3𝑥2

5𝑦4

−4

•

𝑎
5𝑏

−2

Simplify:

𝑧

2
3

5

3

=𝑧

2
3

3

53

= 𝑧2

53

=𝑧2

125

𝑤5

4

3

=𝑤15

43

=𝑤15

64

Simplify:

1. 16

1
4 =

2. 64

1
2 =

3. 25

3
2 =

4. 16

3
4 =

5.
2𝑐

3
5 ∗ 2𝑐

1
5 =

6.

3𝑗

2
3 ∗ 7𝑚

1
4
3𝑗

1
6 ∗ 7𝑚

3
2
=

Raising a Power to a Power

𝑥5 2= 𝑥5∗ 𝑥5= 𝑥5+5= 𝒙𝟏𝟎

 Words: To raise a power to a power, multiply the exponents.

 Algebra: 𝒂𝒎 𝒏= 𝒂𝒎𝒏, where 𝑎 ≠ 0 and m and n are

rational numbers.

Examples

1.

42 3= 42∗ 42∗ _____ =

2.

54 2= 54_____2=

3.
𝑤5 2=

4.

𝑚

1
3

5

= 𝑚

1
3_____5=

𝟒𝟐
𝟒𝟔

∗
𝟓𝟖

𝒘𝟏𝟎

∗
𝒎

𝟓
𝟑