Exponents and Radicals Worksheets | Exponents & Radicals ... - Free Printable

Let’s simplify each radical expression one by one. We’ll factor the radicands (the expressions under the root) into perfect powers and simplify using properties of radicals.

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1) \( \sqrt[3]{448b^7n^2} \)

Factor 448:
\( 448 = 64 \times 7 = 4^3 \times 7 \)

So:
\( \sqrt[3]{448b^7n^2} = \sqrt[3]{64 \cdot 7 \cdot b^7 \cdot n^2} = \sqrt[3]{4^3 \cdot 7 \cdot b^6 \cdot b \cdot n^2} \)

Break into cubes:
= \( \sqrt[3]{4^3} \cdot \sqrt[3]{b^6} \cdot \sqrt[3]{7bn^2} = 4b^2 \sqrt[3]{7bn^2} \)

✔ Answer: \( 4b^2 \sqrt[3]{7bn^2} \)

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2) \( \sqrt{12p} \)

Factor 12:
\( 12 = 4 \cdot 3 \), so:

\( \sqrt{12p} = \sqrt{4 \cdot 3 \cdot p} = \sqrt{4} \cdot \sqrt{3p} = 2\sqrt{3p} \)

✔ Answer: \( 2\sqrt{3p} \)

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3) \( \sqrt{36z^2} \)

This is a perfect square:
\( \sqrt{36z^2} = \sqrt{(6z)^2} = |6z| = 6|z| \)

Since square root yields non-negative result, we use absolute value for variable if necessary.

✔ Answer: \( 6|z| \)

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4) \( \sqrt{8} \)

\( 8 = 4 \cdot 2 \), so:

\( \sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \)

✔ Answer: \( 2\sqrt{2} \)

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5) \( \sqrt[3]{8} \)

\( 8 = 2^3 \), so:

\( \sqrt[3]{8} = \sqrt[3]{2^3} = 2 \)

✔ Answer: \( 2 \)

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6) \( \sqrt[3]{64} \)

\( 64 = 4^3 \), so:

\( \sqrt[3]{64} = 4 \)

✔ Answer: \( 4 \)

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7) \( \sqrt[3]{64z^3} \)

\( 64 = 4^3 \), \( z^3 = (z)^3 \), so:

\( \sqrt[3]{64z^3} = \sqrt[3]{4^3 \cdot z^3} = 4z \)

✔ Answer: \( 4z \)

*(Note: Cube roots of odd powers do not require absolute values since cube root preserves sign.)*

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8) \( \sqrt{112} \)

Factor 112:
\( 112 = 16 \cdot 7 \)

\( \sqrt{112} = \sqrt{16 \cdot 7} = \sqrt{16} \cdot \sqrt{7} = 4\sqrt{7} \)

✔ Answer: \( 4\sqrt{7} \)

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9) \( \sqrt{2750h^5} \)

First, factor 2750:
2750 = 25 × 110 = 25 × 10 × 11 = 25 × 2 × 5 × 11 = 2 × 5³ × 11

So:
\( \sqrt{2750h^5} = \sqrt{2 \cdot 5^3 \cdot 11 \cdot h^5} = \sqrt{5^2 \cdot 5 \cdot 2 \cdot 11 \cdot h^4 \cdot h} \)

= \( \sqrt{5^2} \cdot \sqrt{h^4} \cdot \sqrt{5 \cdot 2 \cdot 11 \cdot h} = 5h^2 \sqrt{110h} \)

✔ Answer: \( 5h^2 \sqrt{110h} \)

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10) \( \sqrt[3]{135y^7p^2} \)

Factor 135:
135 = 27 × 5 = 3³ × 5

So:
\( \sqrt[3]{135y^7p^2} = \sqrt[3]{3^3 \cdot 5 \cdot y^6 \cdot y \cdot p^2} = \sqrt[3]{3^3} \cdot \sqrt[3]{y^6} \cdot \sqrt[3]{5yp^2} \)

= \( 3y^2 \sqrt[3]{5yp^2} \)

✔ Answer: \( 3y^2 \sqrt[3]{5yp^2} \)

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## ✔ Final Answers:

1) \( 4b^2 \sqrt[3]{7bn^2} \)
2) \( 2\sqrt{3p} \)
3) \( 6|z| \)
4) \( 2\sqrt{2} \)
5) \( 2 \)
6) \( 4 \)
7) \( 4z \)
8) \( 4\sqrt{7} \)
9) \( 5h^2 \sqrt{110h} \)
10) \( 3y^2 \sqrt[3]{5yp^2} \)

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💡 Key Concepts Used:

- Perfect squares/cubes: Factor radicand to pull out perfect powers.
- Even-indexed roots (like √): Require absolute value when simplifying variables with even exponents or unknown signs.
- Odd-indexed roots (like ∛): Do NOT require absolute value — they preserve the sign of the radicand.
- Radical rules: \( \sqrt[n]{a^m} = a^{m/n} \), and \( \sqrt[n]{ab} = \sqrt[n]{a} \cdot \sqrt[n]{b} \).

Let me know if you’d like step-by-step breakdowns for any specific problem!