Day 3: Laws of Exponents

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An Introduction to Exponents

Building the Rules Step-by-Step

Let's Start with the Basics

An exponent is just a shortcut for repeated multiplication.

$x^4 = x \cdot x \cdot x \cdot x$

The big number on the bottom is the base.

The small number up top is the exponent.

Rule #1: The Product Rule

When you multiply terms with the same base...

...you ADD the exponents!

Why?

$(x \cdot x) \cdot (x \cdot x \cdot x)$
is just five x's multiplied together!

Product Rule: Example

Simplify: $a^3 \cdot a^5$

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

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

$a^{3+5} = a^8$

Product Rule with Numbers

Simplify: $(4x^2)(5x^6)$

Step 1: Multiply the big numbers (coefficients) first.
$\rightarrow 4 \cdot 5 = 20$

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

Answer:

$20x^8$

Rule #2: The Quotient Rule

When you divide terms with the same base...

...you SUBTRACT the exponents!

Why?

$\frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x}$

allows you to cancel out terms!

Quotient Rule: Example

Simplify: $\frac{b^{10}}{b^4}$

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

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

$b^{10-4} = b^6$

Quotient Rule with Numbers

Simplify: $\frac{12y^9}{3y^2}$

Step 1: Divide the big numbers (coefficients) first.
$\rightarrow 12 \div 3 = 4$

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

Answer:

$4y^7$

Rule #3: The Power Rule

When you raise a power to another power...

...you MULTIPLY the exponents!

Why?

$(x^2)^3$
means
$x^2 \cdot x^2 \cdot x^2$
which is a product rule problem!

Power Rule: Example

Simplify: $(c^5)^4$

1. We are raising a power to another power.

2. We multiply the exponents.

$c^{5 \cdot 4} = c^{20}$

Power Rule with Numbers

Simplify: $(3k^2)^3$

Watch out! The power applies to EVERYTHING inside!

Step 1: Apply the power to the number.
$\rightarrow 3^3 = 27$

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

Answer:

$27k^6$

Let's Review the Rules

Product Rule:

$x^a \cdot x^b = x^{a+b}$

(Same base, ADD exponents)

Quotient Rule:

$\frac{x^a}{x^b} = x^{a-b}$

(Same base, SUBTRACT exponents)

Power Rule:

$(x^a)^b = x^{a \cdot b}$

(Power to a power, MULTIPLY exponents)

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: $(x^3)^4 \cdot x^2$

Exit Ticket

Simplify each expression:

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