c) $$ \sqrt[5]{\frac{1024\left(x^{-1}\right)^{10}}{\left(2 x^{-3}\right)^{5}}} $$

Question

UpStudy AI · Accepted Answer

Let's simplify the expression step by step:

$$
\sqrt[5]{\frac{1024\left(x^{-1}\right)^{10}}{\left(2 x^{-3}\right)^{5}}}
$$

**Step 1: Simplify the exponents**

First, apply the power of a power rule, $$(a^m)^n = a^{m \cdot n}$$:

$$
\left(x^{-1}\right)^{10} = x^{-10}
$$
$$
\left(2 x^{-3}\right)^{5} = 2^{5} \cdot \left(x^{-3}\right)^{5} = 32 x^{-15}
$$

**Step 2: Rewrite the expression with simplified exponents**

$$
\sqrt[5]{\frac{1024 \cdot x^{-10}}{32 \cdot x^{-15}}}
$$

**Step 3: Simplify the coefficients and the variables separately**

Divide the coefficients:

$$
\frac{1024}{32} = 32
$$

For the variables, use the property $$\frac{a^m}{a^n} = a^{m - n}$$:

$$
\frac{x^{-10}}{x^{-15}} = x^{-10 - (-15)} = x^{5}
$$

**Step 4: Combine the simplified coefficients and variables**

$$
\sqrt[5]{32 \cdot x^{5}}
$$

**Step 5: Separate the radical using the product rule for radicals**

$$
\sqrt[5]{32} \cdot \sqrt[5]{x^{5}}
$$

**Step 6: Evaluate each radical**

- $$\sqrt[5]{32}$$: Since $$32 = 2^5$$,
  $$
  \sqrt[5]{32} = 2
  $$
  
- $$\sqrt[5]{x^{5}}$$:
  $$
  \sqrt[5]{x^{5}} = x
  $$

**Step 7: Multiply the results**

$$
2 \cdot x = 2x
$$

**Final Answer:**

$$
2x
$$
The simplified form of the expression is $$2x$$.

c)