Rewriting Radical and Exponential Expressions

Deck 1 of 2: Notation Conversion

In this lesson:

• Convert any radical to exponential form:
• Convert any rational exponent back to radical form
• Handle negative rational exponents

Your Learning Goals for This Unit

By the end, you should be able to:

1. Convert to exponential form:
2. Convert to radical form:
3. Apply product, quotient, and power rules to rational exponents
4. Choose the form that makes computation easier
5. Simplify products and quotients by converting radicals first

When Your Calculator Needs Exponent Form

You need to enter into a calculator app that only accepts exponent notation.

• How do you rewrite as a power?
• What exponent gives the same result as "cube root squared"?

You know from last lesson — how does this generalize?

What You Already Know From Lesson 1

From HSN.RN.A.1:

Logic: "What power of , used twice, gives ?" →

"What power of , used three times, gives ?"

Index Becomes Denominator; Power Becomes Numerator

Converting Radicals: First Three Examples

Example 1:   (index 3 → denom; power 1 → numer)

Example 2:   (index 4 → denom; power 3 → numer)

Example 3:   (index 5 → denom; power 2 → numer)

When the Index Is Missing: Square Root

Why: The symbol has an implied index of 2 — always there, just not written.

Rule: No index written → index is 2.

Rational Exponents Distribute Over Products

Key step: The exponent distributes over a product

This works for products only — not sums:

Check: ✓, so ✓

Quick Check: Convert These Three Radicals

Convert each radical to exponential form:

Write your answers before the next slide.

Find the Error (Misconception 1)

A student writes:

Your task: Find the error. What did the student do wrong? What is the correct answer?

Think before the next slide — where exactly did the 3 and 2 get placed?

Error Resolved: Index Goes to Denominator

Running the Bridge in Reverse

• Denominator → root index
• Numerator → power

Both radical forms are equivalent — use whichever is easier to evaluate.

Exponential to Radical: Three Examples

Example 1:   (denom 3 → index; numer 2 → power)

Example 2:   (denom 4 → index; numer 5 → power)

Example 3:   (denom 2 → square root; numer 1 → no written power)

Compound Base: The Whole Expression

Key: The base is the entire expression — not just .

Compare:

• means — the 2 is a coefficient, not part of the base
• means the entire is raised to the power

Handling Negative Rational Exponents: Two Steps

Rule: Apply the negative exponent rule first, then convert the positive part.

Steps:

1. → reciprocal:
2. Convert →
3. Final answer:

Converting Back: Full Example with Negative Exponent

A student is asked to write in radical form. Work through it:

Step 1: Handle the negative —

Step 2: Convert → denominator 4 gives , numerator 3 gives power 3

Quick Check: Convert These Back to Radicals

Convert each to radical form:

Write your answers — pay attention to which direction the bridge runs here.

Practice: Converting Expressions in Both Directions

Write each step.

Your Turn: No Scaffolding Provided

Convert each:

No hints. Write each answer before advancing.

Summary: The Notation Bridge in Both Directions

• Index → denominator. Power → numerator.
• No index → index is 2. Negative exponent → reciprocal first.

Watch out:

• , not (index → denominator)
• (no distribution over addition)

Coming Up in Deck 2

Deck 2 applies the three exponent rules to rational exponents:

• Multiply like bases:
• Raise to a power:
• Divide like bases:

Same rules as Grade 8 — fraction arithmetic in the exponent is what's new.