POWER PLAY

Mastering the Laws of Exponents

Standard 8.EE.1: Integer Exponents

The Anatomy of a Power

x n

x = Base The number being multiplied

n = Exponent How many times to multiply

Quick Check:

In the expression \( 5^3 \):

What is the base?
What is the exponent?
What does it equal?

Pattern Discovery: Multiplication

\( 2^3 \cdot 2^2 \)

Expanded: \( (2 \cdot 2 \cdot 2) \cdot (2 \cdot 2) \)

Condensed: \( 2^5 \)

\( x^4 \cdot x^2 \)

Expanded: \( (x \cdot x \cdot x \cdot x) \cdot (x \cdot x) \)

Condensed: \( x^6 \)

What is the rule?

When multiplying powers with the same base, what do we do with the exponents?

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

"Keep the base, add the space!"

Pattern Discovery: Division

\( \frac{5^6}{5^2} = \frac{5 \cdot 5 \cdot 5 \cdot 5 \cdot \cancel{5} \cdot \cancel{5}}{\cancel{5} \cdot \cancel{5}} \)

Result: \( 5^4 \)

Exponent Logic: \( 6 - 2 = 4 \)

The Quotient Rule

When dividing powers with the same base, what do we do with the exponents?

\( \frac{x^a}{x^b} = x^{a-b} \)

"Keep the base, subtract the space!"

Pattern Discovery: Power of a Power

\( (3^2)^4 \)

Repeated multiplication:
\( 3^2 \cdot 3^2 \cdot 3^2 \cdot 3^2 \)
\( (3\cdot 3) \cdot (3\cdot 3) \cdot (3\cdot 3) \cdot (3\cdot 3) \)

Condensed: \( 3^8 \)

Power of a Power Rule

When raising a power to another power, what do we do?

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

"Power to a power? Multiply!"

The Zero Exponent Mystery

Quotient Rule:

\( \frac{x^3}{x^3} = x^{3-3} = x^0 \)

Division logic:

\( \frac{x^3}{x^3} = 1 \)

Zero Power Rule

\( x^0 = 1 \)

Anything (except zero) raised to the zero power equals 1.

Negative Exponents

The Elevator Rule

If it's on top, move it to the bottom to make it positive.
If it's on bottom, move it to the top to make it positive.

Negative Rule

\( x^{-n} = \frac{1}{x^n} \)

\( \frac{1}{x^{-n}} = x^n \)

Cross the line, change the sign!

Power Practice

1. \( 7^3 \cdot 7^4 = \) ______

2. \( \frac{x^8}{x^2} = \) ______

3. \( (y^5)^3 = \) ______

4. \( 10^0 = \) ______

5. \( 3^{-2} = \) ______

6. \( \frac{2^2}{2^5} = \) ______

Exponent Power Slides

Exponent Power Slides

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