Simplify ¹²√4x¹⁵ and show the result to be x¹²√4x³

Let's simplify the expression \( \sqrt[12]{4x^{15}} \) step by step:

### Expression:
\[
\sqrt[12]{4x^{15}}
\]

### Step 1: Break the expression under the radical
\[
\sqrt[12]{4x^{15}} = \sqrt[12]{4} \cdot \sqrt[12]{x^{15}}
\]

### Step 2: Simplify \( \sqrt[12]{4} \)
The value of \( \sqrt[12]{4} \) is left as is because 4 is not a perfect 12th power. So:
\[
\sqrt[12]{4}
\]

### Step 3: Simplify \( \sqrt[12]{x^{15}} \)
Use the rule \( \sqrt[n]{x^m} = x^{\frac{m}{n}} \):
\[
\sqrt[12]{x^{15}} = x^{\frac{15}{12}} = x^{\frac{5}{4}}
\]

### Step 4: Combine the results
Now, putting everything together:
\[
\sqrt[12]{4x^{15}} = \sqrt[12]{4} \cdot x^{\frac{5}{4}}
\]

### Step 5: Re-express to match the desired form
The desired result is in the form \( x^3 \cdot \sqrt[12]{4x^3} \), so we rewrite:
\[
x^{\frac{5}{4}} = x^1 \cdot x^{\frac{1}{4}} = x \cdot x^{\frac{1}{4}}
\]
Thus, we have:
\[
x \cdot \sqrt[12]{4x^3}
\]

Therefore, the simplified expression is:
\[
x \cdot \sqrt[12]{4x^3}
\]

This shows the result \( \sqrt[12]{4x^{15}} = x \cdot \sqrt[12]{4x^3} \).

Would you like to dive into more details or ask questions? Here are five related questions to expand your understanding:

1. How would you simplify \( \sqrt[6]{x^{10}} \)?
2. Can you express \( \sqrt[3]{8x^6} \) in a similar form?
3. What is the general rule for simplifying \( \sqrt[n]{x^m} \)?
4. How do fractional exponents work in radicals?
5. How do you simplify the product of two different radicals, like \( \sqrt{2} \cdot \sqrt{3} \)?

**Tip:** Remember, radicals and exponents are deeply interconnected: \( \sqrt[n]{x^m} \) is equivalent to \( x^{m/n} \), which helps simplify many complex expressions.

Master the simplification of radical expressions like ¹²√4x¹⁵, leading to the result x¹²√4x³. Explore key algebraic concepts including exponents and radicals with a step-by-step guide, ideal for Grades 10-12.

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