https://file.aiquickdraw.com/mathbot/file/1726781311686qutfm3ke.jpg

The question asks to simplify the expression \( \left( 16x^8 y^{-12} \right)^{\frac{1}{2}} \).

### Step-by-step solution:

1. **Simplifying the coefficients:**
   \[
   \left( 16 \right)^{\frac{1}{2}} = \sqrt{16} = 4
   \]

2. **Simplifying the exponents of \(x\):**
   Use the exponent rule \( \left( x^a \right)^b = x^{a \cdot b} \):
   \[
   \left( x^8 \right)^{\frac{1}{2}} = x^{8 \cdot \frac{1}{2}} = x^4
   \]

3. **Simplifying the exponents of \(y\):**
   Similarly, apply the same rule to \( y^{-12} \):
   \[
   \left( y^{-12} \right)^{\frac{1}{2}} = y^{-12 \cdot \frac{1}{2}} = y^{-6}
   \]

Now, putting it all together, the expression becomes:
\[
4x^4 y^{-6}
\]

Since \( y^{-6} = \frac{1}{y^6} \), the final expression is:
\[
\frac{4x^4}{y^6}
\]

### Answer:
The correct option is **C: \(\frac{4x^4}{y^6}\)**.

Would you like further explanation or more details? Here are 5 questions to deepen your understanding:

1. Why does \( \sqrt{16} \) simplify to 4?
2. How do exponent rules apply when raising powers to a fraction?
3. What happens when you raise a negative exponent to a fractional power?
4. How would this problem change if the coefficient were not a perfect square?
5. How do you simplify expressions with multiple variables raised to different powers?

**Tip:** When simplifying exponents, always apply exponent rules carefully to both the base and the exponent, ensuring you handle negative exponents properly!

Which expression is equivalent to \( \left( 16x^8 y^{-12} \right)^{\frac{1}{2}} \)?

Learn how to simplify the exponential expression \( \left( 16x^8 y^{-12} \right)^{\frac{1}{2}} \) with a step-by-step solution. This problem involves key algebraic concepts such as exponent rules and simplification techniques. Suitable for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation