Rational Exponents &

Radicals

Mrs. Daniel- Algebra 1

Exponents

Definition: Exponent

• The exponent of a number says how many times to use that number in a multiplication. It is written as a small number to the right and above the base number.

The Zero Exponent Rule

• Any number (excluding zero) to the zero power is always equal to one.

• Examples:

1000=1

1470=1

550 =1

Negative Power Rule

Let’s Practice…1. 5-2

2.3

4

−1

3. (-3)-3

The One Exponent Rule

• Any number (excluding zero) to the first power is always equal to that number.

• Examples:

a1 = a

71 = 7

531 = 53

The Power Rule (Powers to Powers)

When an exponential expression is raised to a power, multiply the exponents.

Try these…

1. (w4)2 2. (q2)8

3. (x3)4

Products to Powers

(ab)n = anbn

Distribute the exponent/power to allvariables and/or coefficients.

For example:

(6y) 2 = (62)(y2)= 36y2

(7x3)2 = (7)2(x3)2 = 49x6

Let’s Practice…1. (5x2)2 4. (6x4)2

2. (3wk3)3 5. (n5)2(4mn-2)3

3. (-2y)4 6. (5x2)2

The Quotient Rule

Let’s Practice…

Power of a Fraction

Let’s Practice…

3. 3

4

−2

Simplifying Radicals

Radical Vocab

How to Simplify Radicals

1. Make a factor tree of the radicand.

2. Circle all final factor pairs.

3. All circled pairs move outside the radical and become single value.

4. Multiply all values outside radical.

5. Multiply all final factors that were not circled. Place product under radical sign.

Let’s Practice…

1. 225 2. 300

Let’s Practice…

3.1

494. 120

How to Simplify Cubed Radicals

1. Make a factor tree of the radicand.

2. Circle all final factor groups of three.

3. All circled groups of three move outside the radical and become single value.

4. Multiply all values outside radical.

5. Multiply all final factors that were not circled. Place product under radical sign.

Let’s Practice…

1. 3375 2.

364

Simplifying Rational

Exponents

Review: Radical Vocab

Code: Fractional Exponents

Rewrite each of the following as a single power of 7:

1. 7

2. 37

3. 73

4. 349

Let’s Practice #1

Rewrite each of the following in radical form:

1. 51

2

2. 51

3

3. 53

2

4. 5−1

4

Let’s Practice #2

Let’s Practice #3

Simplify, if possible:

1. 81

3

2. 161

2 + 271

3

3. 811

4 + 91

2

4. 45

2 - 43

2

Let’s Practice #4

Which is equivalent to a1

2 ∙ b3

4?

A. ab3

B. ab3

C.3a2b4

D.4a2b3

Which is equivalent to 3a2?

A. a3

2

B. a2

3

C. a1

6

D. a6

Let’s Practice #5

Let’s Practice #6

Simplify. Rewrite each of the following as a single power of 7.

1. (491

3 )(7−1

4 )

2. 37

7

3. Rewrite 8 ⋅ 22

5 as a single power of 2.

Let’s Practice #7

Rewrite as radical expressions, then simplify, if possible:

1. 12𝑎2

3

2. 6𝑥5

2

3. 64𝑎4

5

Mini Quiz

Is each statement, true or false. Explain!!!!

Applications

Applications

Applications

The volume of a cube is related to the area of a

face by the formula V = 𝐴3

2. What is the volume of a cube whose face has an area of 100 cm2

Rational & Irrational Numbers

Rational Numbers• Any number that can be expressed as the

quotient or fraction 𝑝

𝑞of two integers.

• YES:

– Any integers

– Any decimals that

ends or repeats

– Any fraction

• NO:

– Never ending decimals

Irrational Numbers

• Any number that can not be expressed as a fraction.

• Usually a never-ending, non-repeating decimal.

• Examples:

𝜋

2, 5

1.2658945625692….

Let’s Practice…

Rational or Irrational.

1.2

17

2.1

3

3. 0

Will it be Rational or Irrational?

Sums:

Rational + Rational =

Rational + Irrational =

Irrational + Irrational =

Products:

Rational x Rational =

Rational x Irrational =

Irrational x Irrational =

Rational or Irrational?

1. Is the sum of 3 2 and 4 2 rational or irrational?

2. Is the sum of 4.2 and 2 rational or irrational?

3. Determine if the product of 3 2 and 8 18is rational or irrational.

Advanced Rational

Expressions