Simplest form of `root(4r)(x^6)+root(2r)(z^(-5))` is

Options

A. `root4(z^r)`
B. `rootr(z^4)`
C. `sqrt(z^(4r))`
D. none of the above

Answer

To simplify the expression \( \sqrt{4r} \cdot x^6 + \sqrt{2r} \cdot z^{-5} \), we will follow these steps:

### Step 1: Simplify \( \sqrt{4r} \cdot x^6 \)
We can break this down into two parts:
- The square root of \( 4r \)
- The term \( x^6 \)

The square root of \( 4r \) can be simplified as:
\[ \sqrt{4r} = \sqrt{4} \cdot \sqrt{r} = 2\sqrt{r} \]

Now, we can rewrite \( x^6 \) as:
\[ x^6 = (x^3)^2 \]
Thus, we can express \( \sqrt{x^6} \) as:
\[ \sqrt{x^6} = x^3 \]

Combining these, we have:
\[ \sqrt{4r} \cdot x^6 = 2\sqrt{r} \cdot x^3 = 2x^3\sqrt{r} \]

### Step 2: Simplify \( \sqrt{2r} \cdot z^{-5} \)
Similarly, we can simplify \( \sqrt{2r} \):
\[ \sqrt{2r} = \sqrt{2} \cdot \sqrt{r} \]
And for \( z^{-5} \), we can express it as:
\[ \sqrt{z^{-5}} = z^{-5/2} \]

Thus, we can combine these:
\[ \sqrt{2r} \cdot z^{-5} = \sqrt{2} \cdot \sqrt{r} \cdot z^{-5} = \sqrt{2} \cdot z^{-5/2} \cdot \sqrt{r} \]

### Step 3: Combine the two simplified expressions
Now we can combine the two parts:
\[ 2x^3\sqrt{r} + \sqrt{2}z^{-5/2}\sqrt{r} \]

Factoring out \( \sqrt{r} \):
\[ \sqrt{r}(2x^3 + \sqrt{2}z^{-5/2}) \]

### Final Answer
The simplest form of the expression \( \sqrt{4r} \cdot x^6 + \sqrt{2r} \cdot z^{-5} \) is:
\[ \sqrt{r}(2x^3 + \sqrt{2}z^{-5/2}) \]

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