Algebra 1 // Unit 4

Radical Architect

Building mastery in simplifying radical expressions with exponents and prime factorization.

Warm-Up: Power Play

Review your exponent rules. How do we combine powers with the same base?

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

Try These:

1) \(x^3 \cdot x^2 = \_\_\_\_\)

2) \(9^2 \cdot 9^1 = \_\_\_\_\)

3) \(y^5 \cdot y^? = y^{12}\)

The Factor Strategy

Watch closely at Example 3:

The narrator encounters \(9^3\) inside a radical. Since we are finding the square root, we need even powers.

\(9^3 = 9^2 \cdot 9^1\)

Why did he do this? What happens to the \(9^2\)?

Mastering Large Radicals

Focus: 5:15 - 5:45

Split and Root

To simplify a variable with an odd power:

1 Split the power into the largest even number and 1.
2 Take the square root of the even power (divide exponent by 2).
3 Keep the remaining power inside the radical.

Example: \(\sqrt{x^5}\)

\(\sqrt{x^4 \cdot x^1}\)

Simplify

\(x^2 \sqrt{x}\)

Activity: Radical Puzzle

The Challenge:

Your blueprint is missing its dimensions! You must simplify 8 radical expressions to unlock the building's final height.

"Remember: Large numbers can be broken into perfect squares you already know, like \(16\), \(81\), or \(361\)!"

\(\sqrt{2^7}\)

\(\sqrt{x^9}\)

\(\sqrt{12x^3}\)

???

The Golden Rule

For Square Roots:

To bring a variable outside,
DIVIDE the exponent by 2.

\(\sqrt{x^n} = x^{n/2}\)

\(\sqrt{x^{20}} = x^{10}\)

\(\sqrt{x^{11}} = x^{5}\sqrt{x}\)

Architect's Bonus

Cube Root Expansion

If square roots require dividing by 2, what would a cube root (\(\sqrt[3]{x}\)) require?

Think about how you would handle \(\sqrt[3]{x^7}\). How would you split it?

Draft your rule:

\(\sqrt[3]{x^n} = \dots\)

Radical Architect

Radical Architect Slides

Radical Architect Worksheet

Radical Architect Answer Key

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