Worksheet for reducing square and cube roots with multiple-choice questions.

Let's solve each of the first three problems step by step, as they are about reducing radicals (square and cube roots). We'll focus on simplifying expressions under the radical sign by factoring out perfect powers.

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Problem 1:

Reduce the radical
$$
\sqrt[3]{250x^5}
$$

#### Step 1: Factor the coefficient and variable
- $ 250 = 2 \times 125 = 2 \times 5^3 $
- $ x^5 = x^3 \cdot x^2 $

So:
$$
\sqrt[3]{250x^5} = \sqrt[3]{2 \cdot 5^3 \cdot x^3 \cdot x^2}
$$

Now take out perfect cubes:
- $ \sqrt[3]{5^3} = 5 $
- $ \sqrt[3]{x^3} = x $
- Remaining: $ \sqrt[3]{2x^2} $

So:
$$
\sqrt[3]{250x^5} = 5x \cdot \sqrt[3]{2x^2}
$$

✔ Answer: A) $ 5x \sqrt[3]{2x^2} $

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Problem 2:

Reduce the radical
$$
\sqrt[3]{135m^8}
$$

#### Step 1: Factor 135 and $ m^8 $
- $ 135 = 27 \times 5 = 3^3 \times 5 $
- $ m^8 = m^6 \cdot m^2 = (m^3)^2 \cdot m^2 $

So:
$$
\sqrt[3]{135m^8} = \sqrt[3]{3^3 \cdot 5 \cdot m^6 \cdot m^2}
$$

Take out perfect cubes:
- $ \sqrt[3]{3^3} = 3 $
- $ \sqrt[3]{m^6} = m^2 $ (since $ m^6 = (m^2)^3 $)
- Remaining: $ \sqrt[3]{5m^2} $

So:
$$
\sqrt[3]{135m^8} = 3m^2 \cdot \sqrt[3]{5m^2}
$$

✔ Answer: B) $ 3m^2 \sqrt[3]{5m^2} $

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Problem 3:

Reduce the radical
$$
\sqrt{108w^3}
$$

Note: This is a square root, not cube root.

#### Step 1: Factor 108 and $ w^3 $
- $ 108 = 36 \times 3 = 6^2 \times 3 $
- $ w^3 = w^2 \cdot w $

So:
$$
\sqrt{108w^3} = \sqrt{6^2 \cdot 3 \cdot w^2 \cdot w}
$$

Take out perfect squares:
- $ \sqrt{6^2} = 6 $
- $ \sqrt{w^2} = w $
- Remaining: $ \sqrt{3w} $

So:
$$
\sqrt{108w^3} = 6w \cdot \sqrt{3w}
$$

✔ Answer: D) $ 6w \sqrt{3w} $

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✔ Final Answers:

1. A
2. B
3. D

Let me know if you'd like help with the rest!