Lesson 4.11

General Rational Exponents

Now we combine both ideas: means "take the nth root, then raise to the mth power" (or vice versa). The fraction exponent packs two operations into one notation.

Introduction

From 4.10, . Now extend: means apply both a root and a power. You can do them in either order — root first is usually easier.

Past Knowledge

Unit fraction exponents (4.10), exponent rule .

Today's Goal

Evaluate and simplify expressions with .

Future Success

4.12 covers fluently switching between radical and exponential forms.

Key Concepts

Two Equivalent Forms

Power first

Root first ✓

Root first keeps numbers smaller!

Reading the Fraction

📐 Numerator (m) = the Power

🌱 Denominator (n) = the Root

"Power over Root" — or think "the flower is on top, the root is below"

Negative Rational Exponents

The negative sign means reciprocal — same rule as integer exponents!

Worked Examples

Example 1: Evaluate

Basic

Evaluate .

1

Root first: cube root of 8

2

Then raise to the 2nd power

Example 2: Larger Numbers

Intermediate

Evaluate .

1

Root first: fourth root of 16

2

Raise to the 3rd power

Example 3: Negative Exponent

Advanced

Evaluate .

1

Handle the negative: take reciprocal

2

Evaluate

Common Pitfalls

Power First = Huge Numbers

: doing then works but is harder. Root first: .

Numerator/Denominator Mix-up

In , the 3 is the root (denominator = down = root in the ground), the 2 is the power (numerator = up).

Real-Life Applications

Kleiber's Law states that an animal's metabolic rate scales as where is body mass in kg. This 3/4 power law governs biology from mice to elephants.

Practice Quiz

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