Given x, is greater than, 0, comma*x*>0, express in simplest radical form with a rational denominator.

start fraction, square root of, 32, x, to the power 12 , end square root, divided by, square root of, 2, x, squared, end square root, end fraction

2*x*2​32*x*12​​

We are given the following expression:

\[
\frac{\sqrt{32x^{12}}}{\sqrt{2x^2}}
\]

### Step 1: Simplify the numerator
The numerator is \(\sqrt{32x^{12}}\). We can break this into:

\[
\sqrt{32} \cdot \sqrt{x^{12}}
\]

We know that \(\sqrt{32} = \sqrt{16 \cdot 2} = 4\sqrt{2}\) and \(\sqrt{x^{12}} = x^6\). So, the numerator becomes:

\[
4\sqrt{2} \cdot x^6 = 4x^6\sqrt{2}
\]

### Step 2: Simplify the denominator
The denominator is \(\sqrt{2x^2}\), which can be split into:

\[
\sqrt{2} \cdot \sqrt{x^2}
\]

Since \(\sqrt{x^2} = x\), the denominator becomes:

\[
\sqrt{2} \cdot x = x\sqrt{2}
\]

### Step 3: Combine numerator and denominator
Now, we can simplify the full expression:

\[
\frac{4x^6\sqrt{2}}{x\sqrt{2}}
\]

### Step 4: Cancel common terms
We can cancel out the \(\sqrt{2}\) terms and simplify \(x^6 / x = x^5\):

\[
\frac{4x^6\sqrt{2}}{x\sqrt{2}} = 4x^5
\]

Thus, the simplified form of the expression is:

\[
4x^5
\]

Let me know if you'd like any further explanation!

Here are five related questions to explore:

1. How do you simplify expressions involving radicals with variables?
2. What are the properties of exponents that help in simplifying radicals?
3. How can you rationalize the denominator in radical expressions?
4. Why does \(\sqrt{x^2} = x\) for \(x > 0\)?
5. How would this change if the variables were raised to negative exponents?

**Tip:** When simplifying radical expressions, always look for opportunities to break them into smaller parts, making use of perfect squares and cube roots.

Given x > 0, express in simplest radical form with a rational denominator: start fraction, square root of, 32, x, to the power 12 , end square root, divided by, square root of, 2, x, squared, end square root.

Learn how to simplify the radical expression √(32x^12) / √(2x^2) step-by-step. This guide covers key mathematical concepts such as radicals, exponent rules, and simplifying expressions, suitable for students in Grades 9-12.

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