Simplifying Radical Expressions worksheet with ten problems requiring simplification using absolute value signs if necessary.

Problem: Simplify each radical expression. Use absolute value signs if necessary.

We will simplify each expression step by step. Let's go through them one by one.

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#### 1) \( \sqrt[6]{64x^5y^8} \)

- Step 1: Express the radicand in terms of its prime factors.
\[
64 = 2^6, \quad x^5 = x^5, \quad y^8 = y^8
\]
So,
\[
\sqrt[6]{64x^5y^8} = \sqrt[6]{2^6 \cdot x^5 \cdot y^8}
\]

- Step 2: Apply the property of radicals: \( \sqrt[n]{a \cdot b} = \sqrt[n]{a} \cdot \sqrt[n]{b} \).
\[
\sqrt[6]{2^6 \cdot x^5 \cdot y^8} = \sqrt[6]{2^6} \cdot \sqrt[6]{x^5} \cdot \sqrt[6]{y^8}
\]

- Step 3: Simplify each term.
- \( \sqrt[6]{2^6} = 2 \) because \( (2^6)^{1/6} = 2 \).
- \( \sqrt[6]{x^5} = x^{5/6} \).
- \( \sqrt[6]{y^8} = y^{8/6} = y^{4/3} \).

- Step 4: Combine the results.
\[
\sqrt[6]{64x^5y^8} = 2 \cdot x^{5/6} \cdot y^{4/3}
\]

- Final Answer:
\[
\boxed{2x^{5/6}y^{4/3}}
\]

---

#### 2) \( \sqrt[7]{176} \)

- Step 1: Factorize 176 into its prime factors.
\[
176 = 2^4 \cdot 11
\]

- Step 2: Write the expression with the prime factorization.
\[
\sqrt[7]{176} = \sqrt[7]{2^4 \cdot 11}
\]

- Step 3: Since none of the exponents are multiples of 7, this expression cannot be simplified further.
\[
\sqrt[7]{176} = \sqrt[7]{2^4 \cdot 11}
\]

- Final Answer:
\[
\boxed{\sqrt[7]{176}}
\]

---

#### 3) \( \sqrt[4]{1500x^4} \)

- Step 1: Factorize 1500 into its prime factors.
\[
1500 = 2^2 \cdot 3 \cdot 5^3
\]

- Step 2: Write the expression with the prime factorization.
\[
\sqrt[4]{1500x^4} = \sqrt[4]{2^2 \cdot 3 \cdot 5^3 \cdot x^4}
\]

- Step 3: Separate the terms that can be simplified.
- \( \sqrt[4]{x^4} = x \) because \( (x^4)^{1/4} = x \).
- For the constants, we cannot simplify \( 2^2 \), \( 3 \), or \( 5^3 \) further under the fourth root.

- Step 4: Combine the results.
\[
\sqrt[4]{1500x^4} = x \cdot \sqrt[4]{2^2 \cdot 3 \cdot 5^3}
\]

- Final Answer:
\[
\boxed{x\sqrt[4]{1500}}
\]

---

#### 4) \( \sqrt[4]{567} \)

- Step 1: Factorize 567 into its prime factors.
\[
567 = 3^4 \cdot 7
\]

- Step 2: Write the expression with the prime factorization.
\[
\sqrt[4]{567} = \sqrt[4]{3^4 \cdot 7}
\]

- Step 3: Simplify the term with the exponent that is a multiple of 4.
- \( \sqrt[4]{3^4} = 3 \).
- The term \( \sqrt[4]{7} \) remains as it is.

- Step 4: Combine the results.
\[
\sqrt[4]{567} = 3 \cdot \sqrt[4]{7}
\]

- Final Answer:
\[
\boxed{3\sqrt[4]{7}}
\]

---

#### 5) \( \sqrt[4]{4x^3} \)

- Step 1: Factorize 4 into its prime factors.
\[
4 = 2^2
\]

- Step 2: Write the expression with the prime factorization.
\[
\sqrt[4]{4x^3} = \sqrt[4]{2^2 \cdot x^3}
\]

- Step 3: Separate the terms.
- \( \sqrt[4]{2^2} = 2^{2/4} = 2^{1/2} = \sqrt{2} \).
- \( \sqrt[4]{x^3} = x^{3/4} \).

- Step 4: Combine the results.
\[
\sqrt[4]{4x^3} = \sqrt{2} \cdot x^{3/4}
\]

- Final Answer:
\[
\boxed{\sqrt{2}x^{3/4}}
\]

---

#### 6) \( \sqrt[6]{960} \)

- Step 1: Factorize 960 into its prime factors.
\[
960 = 2^6 \cdot 3 \cdot 5
\]

- Step 2: Write the expression with the prime factorization.
\[
\sqrt[6]{960} = \sqrt[6]{2^6 \cdot 3 \cdot 5}
\]

- Step 3: Simplify the term with the exponent that is a multiple of 6.
- \( \sqrt[6]{2^6} = 2 \).
- The terms \( \sqrt[6]{3} \) and \( \sqrt[6]{5} \) remain as they are.

- Step 4: Combine the results.
\[
\sqrt[6]{960} = 2 \cdot \sqrt[6]{3 \cdot 5} = 2 \cdot \sqrt[6]{15}
\]

- Final Answer:
\[
\boxed{2\sqrt[6]{15}}
\]

---

#### 7) \( \sqrt[4]{81} \)

- Step 1: Factorize 81 into its prime factors.
\[
81 = 3^4
\]

- Step 2: Write the expression with the prime factorization.
\[
\sqrt[4]{81} = \sqrt[4]{3^4}
\]

- Step 3: Simplify the term.
- \( \sqrt[4]{3^4} = 3 \).

- Final Answer:
\[
\boxed{3}
\]

---

#### 8) \( \sqrt[4]{2816x^7y^3} \)

- Step 1: Factorize 2816 into its prime factors.
\[
2816 = 2^8 \cdot 11
\]

- Step 2: Write the expression with the prime factorization.
\[
\sqrt[4]{2816x^7y^3} = \sqrt[4]{2^8 \cdot 11 \cdot x^7 \cdot y^3}
\]

- Step 3: Separate the terms that can be simplified.
- \( \sqrt[4]{2^8} = 2^{8/4} = 2^2 = 4 \).
- \( \sqrt[4]{x^7} = x^{7/4} \).
- \( \sqrt[4]{y^3} = y^{3/4} \).
- The term \( \sqrt[4]{11} \) remains as it is.

- Step 4: Combine the results.
\[
\sqrt[4]{2816x^7y^3} = 4 \cdot x^{7/4} \cdot y^{3/4} \cdot \sqrt[4]{11}
\]

- Final Answer:
\[
\boxed{4x^{7/4}y^{3/4}\sqrt[4]{11}}
\]

---

#### 9) \( \sqrt[3]{24} \)

- Step 1: Factorize 24 into its prime factors.
\[
24 = 2^3 \cdot 3
\]

- Step 2: Write the expression with the prime factorization.
\[
\sqrt[3]{24} = \sqrt[3]{2^3 \cdot 3}
\]

- Step 3: Simplify the term with the exponent that is a multiple of 3.
- \( \sqrt[3]{2^3} = 2 \).
- The term \( \sqrt[3]{3} \) remains as it is.

- Step 4: Combine the results.
\[
\sqrt[3]{24} = 2 \cdot \sqrt[3]{3}
\]

- Final Answer:
\[
\boxed{2\sqrt[3]{3}}
\]

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#### 10) \( \sqrt{1960} \)

- Step 1: Factorize 1960 into its prime factors.
\[
1960 = 2^3 \cdot 5 \cdot 7^2
\]

- Step 2: Write the expression with the prime factorization.
\[
\sqrt{1960} = \sqrt{2^3 \cdot 5 \cdot 7^2}
\]

- Step 3: Separate the terms that can be simplified.
- \( \sqrt{7^2} = 7 \).
- \( \sqrt{2^3} = \sqrt{2^2 \cdot 2} = 2\sqrt{2} \).
- The term \( \sqrt{5} \) remains as it is.

- Step 4: Combine the results.
\[
\sqrt{1960} = 7 \cdot 2 \cdot \sqrt{2} \cdot \sqrt{5} = 14\sqrt{10}
\]

- Final Answer:
\[
\boxed{14\sqrt{10}}
\]

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Final Answers:

1. \( \boxed{2x^{5/6}y^{4/3}} \)
2. \( \boxed{\sqrt[7]{176}} \)
3. \( \boxed{x\sqrt[4]{1500}} \)
4. \( \boxed{3\sqrt[4]{7}} \)
5. \( \boxed{\sqrt{2}x^{3/4}} \)
6. \( \boxed{2\sqrt[6]{15}} \)
7. \( \boxed{3} \)
8. \( \boxed{4x^{7/4}y^{3/4}\sqrt[4]{11}} \)
9. \( \boxed{2\sqrt[3]{3}} \)
10. \( \boxed{14\sqrt{10}} \)