Which expression is equivalent to $$ 81^{\frac{1}{3}} $$ ? $$ 3 \sqrt[3]{3} $$ $$ 3 \sqrt{3^{3}} $$ $$ 9 \sqrt[3]{3} $$ $$ 27 \sqrt[3]{3} $$

Question

UpStudy AI · Accepted Answer

To determine which expression is equivalent to $$ 81^{\frac{1}{3}} $$, let's break it down step by step.

1. **Express 81 as a Power of 3:**
   $$
   81 = 3^4
   $$

2. **Apply the Exponent:**
   $$
   81^{\frac{1}{3}} = (3^4)^{\frac{1}{3}} = 3^{\frac{4}{3}}
   $$

3. **Simplify the Exponent:**
   $$
   3^{\frac{4}{3}} = 3^{1 + \frac{1}{3}} = 3^1 \times 3^{\frac{1}{3}} = 3 \times \sqrt[3]{3}
   $$

So, $$ 81^{\frac{1}{3}} $$ simplifies to $$ 3 \sqrt[3]{3} $$.

**Verification of Other Options:**

- **Option 2:** $$ 3 \sqrt{3^{3}} = 3 \times 3^{\frac{3}{2}} = 3^{2.5} $$  
  This does not simplify to $$ 3 \sqrt[3]{3} $$.

- **Option 3:** $$ 9 \sqrt[3]{3} = 9 \times 3^{\frac{1}{3}} = 3^{2 + \frac{1}{3}} $$  
  This also does not match $$ 3^{\frac{4}{3}} $$.

- **Option 4:** $$ 27 \sqrt[3]{3} = 27 \times 3^{\frac{1}{3}} = 3^{3 + \frac{1}{3}} $$  
  This is significantly larger than $$ 3^{\frac{4}{3}} $$.

**Conclusion:**

The correct expression equivalent to $$ 81^{\frac{1}{3}} $$ is:

**$$ 3 \sqrt[3]{3} $$**

**Answer:** 
$$ 3 \sqrt[3]{3} $$
$$ 3 \sqrt[3]{3} $$

Which expression is equivalent to \( 81^{\frac{1}{3}} \) ? \( 3 \sqrt[3]{3} \) \( 3 \sqrt{3^{3}} \) \( 9 \sqrt[3]{3} \) \( 27 \sqrt[3]{3} \)