1) $$ \sqrt[5]{32^{4}} $$ 2) $$ \sqrt[4]{64^{5}} $$ 3) $$ \sqrt[4]{81^{-3}} $$

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UpStudy AI · Accepted Answer

Let's simplify each of the given expressions step by step.

### 1) $$ \sqrt[5]{32^{4}} $$

- **Step 1:** Express $$ 32 $$ as a power of 2:
  $$
  32 = 2^5
  $$
- **Step 2:** Raise 32 to the 4th power:
  $$
  32^4 = (2^5)^4 = 2^{20}
  $$
- **Step 3:** Take the 5th root of $$ 2^{20} $$:
  $$
  \sqrt[5]{2^{20}} = 2^{20/5} = 2^4 = 16
  $$

**Answer:** $$ 16 $$

---

### 2) $$ \sqrt[4]{64^{5}} $$

- **Step 1:** Express $$ 64 $$ as a power of 2:
  $$
  64 = 2^6
  $$
- **Step 2:** Raise 64 to the 5th power:
  $$
  64^5 = (2^6)^5 = 2^{30}
  $$
- **Step 3:** Take the 4th root of $$ 2^{30} $$:
  $$
  \sqrt[4]{2^{30}} = 2^{30/4} = 2^{7.5} = 2^7 \times 2^{0.5} = 128 \sqrt{2}
  $$

**Answer:** $$ 128\sqrt{2} $$

---

### 3) $$ \sqrt[4]{81^{-3}} $$

- **Step 1:** Express $$ 81 $$ as a power of 3:
  $$
  81 = 3^4
  $$
- **Step 2:** Apply the exponent:
  $$
  81^{-3} = (3^4)^{-3} = 3^{-12}
  $$
- **Step 3:** Take the 4th root of $$ 3^{-12} $$:
  $$
  \sqrt[4]{3^{-12}} = 3^{-12/4} = 3^{-3} = \frac{1}{27}
  $$

**Answer:** $$ \frac{1}{27} $$

---

**Summary of Answers:**
1. $$ 16 $$
2. $$ 128\sqrt{2} $$
3. $$ \dfrac{1}{27} $$
1. $$ 16 $$ 2. $$ 128\sqrt{2} $$ 3. $$ \frac{1}{27} $$

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