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An Introduction to Exponents

Building the Rules Step-by-Step

Slide 2

Let's Start with the Basics

An exponent is just a shortcut for

repeated multiplication.

$\color{#3b82f6}{x}\color{#ef4444}{^4} = \color{#3b82f6}{x} \cdot \color{#3b82f6}{x} \cdot \color{#3b82f6}{x} \cdot \color{#3b82f6}{x}$

The big number on the bottom is the

base.

The small number up top is the exponent.

Slide 3

Rule #1: The Product Rule

When you multiply terms with the

same base...

...you ADD the exponents!

Why?

$(\color{#3b82f6}{x} \cdot \color{#3b82f6}{x}) \cdot (\color{#3b82f6}{x} \cdot \color{#3b82f6}{x} \cdot \color{#3b82f6}{x})$

is just five x's multiplied together!

Slide 4

Product Rule: Example

Simplify: $\color{#3b82f6}{a}\color{#ef4444}{^3} \cdot \color{#3b82f6}{a}\color{#ef4444}{^5}$

1. The base is the same ('a').

2. We are multiplying, so we add the exponents.


$\color{#3b82f6}{a}\color{#ef4444}{^{3+5}} = \color{#3b82f6}{a}\color{#ef4444}{^8}$

Slide 5

Product Rule with Numbers

Simplify: $(\color{#16a34a}{4}\color{#3b82f6}{x}\color{#ef4444}{^2})(\color{#16a34a}{5}\color{#3b82f6}{x}\color{#ef4444}{^6})$

Step 1: Multiply the coefficients.
$\rightarrow \color{#16a34a}{4} \cdot \color{#16a34a}{5} = \color{#16a34a}{20}$

Step 2: Add the exponents of the variables.
$\rightarrow \color{#3b82f6}{x}\color{#ef4444}{^{2+6}} = \color{#3b82f6}{x}\color{#ef4444}{^8}$

Answer:

$\color{#16a34a}{20}\color{#3b82f6}{x}\color{#ef4444}{^8}$

Slide 6

Rule #2: The Quotient Rule

When you divide terms with the

same base...

...you SUBTRACT the exponents!

Why?

$\frac{\color{#3b82f6}{x} \cdot \color{#3b82f6}{x} \cdot \color{#3b82f6}{x} \cdot \color{#3b82f6}{x} \cdot \color{#3b82f6}{x}}{\color{#3b82f6}{x} \cdot \color{#3b82f6}{x}}$

allows you to cancel out terms!

Slide 7

Quotient Rule: Example

Simplify: $\frac{\color{#3b82f6}{b}\color{#ef4444}{^{10}}}{\color{#3b82f6}{b}\color{#ef4444}{^4}}$

1. The base is the same ('b').

2. We are dividing, so we subtract the exponents.

$\color{#3b82f6}{b}\color{#ef4444}{^{10-4}} = \color{#3b82f6}{b}\color{#ef4444}{^6}$

Slide 8

Quotient Rule with Numbers

Simplify: $\frac{\color{#16a34a}{12}\color{#3b82f6}{y}\color{#ef4444}{^9}}{\color{#16a34a}{3}\color{#3b82f6}{y}\color{#ef4444}{^2}}$

Step 1: Divide the coefficients.
$\rightarrow \color{#16a34a}{12} \div \color{#16a34a}{3} = \color{#16a34a}{4}$

Step 2: Subtract the exponents of the variables.
$\rightarrow \color{#3b82f6}{y}\color{#ef4444}{^{9-2}} = \color{#3b82f6}{y}\color{#ef4444}{^7}$

Answer:

$\color{#16a34a}{4}\color{#3b82f6}{y}\color{#ef4444}{^7}$

Slide 9

Rule #3: The Power Rule

When you raise a power to

another power...

...you MULTIPLY the exponents!

Why?

$(\color{#3b82f6}{x}\color{#ef4444}{^2})\color{#ef4444}{^3}$

means

$\color{#3b82f6}{x}\color{#ef4444}{^2} \cdot \color{#3b82f6}{x}\color{#ef4444}{^2} \cdot \color{#3b82f6}{x}\color{#ef4444}{^2}$

Slide 10

Power Rule: Example

Simplify: $(\color{#3b82f6}{c}\color{#ef4444}{^5})\color{#ef4444}{^4}$

1. We are raising a power to another power.

2. We multiply the exponents.

$\color{#3b82f6}{c}\color{#ef4444}{^{5 \cdot 4}} = \color{#3b82f6}{c}\color{#ef4444}{^{20}}$

Slide 11

Power Rule with Numbers

Simplify: $(\color{#16a34a}{3}\color{#3b82f6}{k}\color{#ef4444}{^2})\color{#ef4444}{^3}$

Watch out! The power applies to EVERYTHING inside!

Step 1: Apply the power to the number.
$\rightarrow \color{#16a34a}{3}\color{#ef4444}{^3} = \color{#16a34a}{27}$

Step 2: Multiply the exponents of the variable.
$\rightarrow \color{#3b82f6}{k}\color{#ef4444}{^{2 \cdot 3}} = \color{#3b82f6}{k}\color{#ef4444}{^6}$

Answer:

$\color{#16a34a}{27}\color{#3b82f6}{k}\color{#ef4444}{^6}$

Slide 12

Let's Review the Rules

Product Rule:

$\color{#3b82f6}{x}\color{#ef4444}{^a} \cdot \color{#3b82f6}{x}\color{#ef4444}{^b} = \color{#3b82f6}{x}\color{#ef4444}{^{a+b}}$

(Same base, ADD exponents)

Quotient Rule:

$\frac{\color{#3b82f6}{x}\color{#ef4444}{^a}}{\color{#3b82f6}{x}\color{#ef4444}{^b}} = \color{#3b82f6}{x}\color{#ef4444}{^{a-b}}$

(Same base, SUBTRACT exponents)

Power Rule:

$(\color{#3b82f6}{x}\color{#ef4444}{^a})\color{#ef4444}{^b} = \color{#3b82f6}{x}\color{#ef4444}{^{a \cdot b}}$

(Power to a power, MULTIPLY exponents)

Slide 13

Time to Practice!

Choose your level and simplify each expression.

LEVEL 1

  1. Simplify: $x^7 \cdot x^3$
  2. Simplify: $y^8 / y^2$
  3. Simplify: $(a^4)^5$

LEVEL 2

  1. Simplify: $(3n^4)(6n^2)$
  2. Simplify: $15b^8 / 3b^5$
  3. Simplify: $(4c^2)^2$

LEVEL 3

  1. Simplify: $(2x^5y^2)(4x^2y)$
  2. Simplify: $(12a^5b^3) / (6ab)$
  3. Simplify: $(x^3)^4 \cdot x^2$
Slide 14

Exit Ticket

Simplify each expression:

  1. $(6k^2)(3k^8)$
  2. $24m^9 / 4m^3$
  3. $(p^6)^3$