50+ cube roots worksheets on Quizizz | Free & Printable - Free Printable

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

#### Step 2: Take out perfect cubes
- $ \sqrt[3]{5^3} = 5 $
- $ \sqrt[3]{x^3} = x $
- Remaining inside: $ \sqrt[3]{2x^2} $

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

✔ Answer: A) $ 5x \sqrt[3]{2x^2} $

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Problem 2:

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

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

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

#### Step 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:
$$
= 3m^2 \cdot \sqrt[3]{5m^2}
$$

✔ 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 = 36 \times 3 = 6^2 \times 3 $
- $ w^3 = w^2 \cdot w $

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

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

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

✔ 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 of the quiz!