Simplify the following expression in terms of fractional exponents and write it in the form 10 to the power of a x to the power of b.

fifth root of 10 to the power of 4 x end root

To simplify the expression \(\sqrt[5]{10^4 \cdot x}\) and write it in the form \(10^a \cdot x^b\), follow these steps:

1. **Break down the expression inside the fifth root:**
   \[
   \sqrt[5]{10^4 \cdot x}
   \]

2. **Apply the property of roots to split the expression:**
   \[
   \sqrt[5]{10^4 \cdot x} = \sqrt[5]{10^4} \cdot \sqrt[5]{x}
   \]

3. **Convert the fifth root into fractional exponents:**
   \[
   \sqrt[5]{10^4} = 10^{\frac{4}{5}}
   \]
   \[
   \sqrt[5]{x} = x^{\frac{1}{5}}
   \]

4. **Combine the simplified terms:**
   \[
   10^{\frac{4}{5}} \cdot x^{\frac{1}{5}}
   \]

Thus, the expression simplified in the form \(10^a \cdot x^b\) is:
\[
10^{\frac{4}{5}} \cdot x^{\frac{1}{5}}
\]

Here, \(a = \frac{4}{5}\) and \(b = \frac{1}{5}\).

Would you like more details or have any questions?

### Related Questions:
1. How do you simplify \(\sqrt[3]{1000 \cdot x^3}\) in a similar form?
2. What is the general rule for converting roots into fractional exponents?
3. How would you simplify \(\sqrt[4]{16 \cdot y^2}\) in terms of fractional exponents?
4. What are the rules for multiplying exponents with the same base?
5. Can you apply this simplification method to logarithmic expressions?

**Tip:** When dealing with roots and exponents, always remember that a root is the same as a fractional exponent, where the numerator is the power and the denominator is the root.

Simplify the following expression in terms of fractional exponents and write it in the form 10 to the power of a x to the power of b: fifth root of 10 to the power of 4 x end root.

Learn how to simplify the fifth root of 10^4 * x and express it in terms of fractional exponents. This problem covers key algebraic concepts, including the conversion of roots to fractional exponents and exponent multiplication. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation