Math Problem Statement

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

Solution

To simplify the given expression, let's follow these steps carefully:

The given expression is:

104x5\sqrt[5]{10^4 x}

Step 1: Express the fifth root as a fractional exponent

The nn-th root of a number can be expressed as a fractional exponent x1/nx^{1/n}. Therefore:

104x5=(104x)1/5\sqrt[5]{10^4 x} = (10^4 x)^{1/5}

Step 2: Distribute the fractional exponent

Using the rule (ab)c=acbc(a \cdot b)^c = a^c \cdot b^c, we can split the expression:

(104x)1/5=10415x15(10^4 x)^{1/5} = 10^{4 \cdot \frac{1}{5}} \cdot x^{\frac{1}{5}}

Step 3: Simplify the exponents

  • For 1041510^{4 \cdot \frac{1}{5}}, simplify the exponent: 415=454 \cdot \frac{1}{5} = \frac{4}{5}.
  • For x15x^{\frac{1}{5}}, the exponent is already simplified.

Thus, the expression becomes:

1045x1510^{\frac{4}{5}} \cdot x^{\frac{1}{5}}

Final Answer:

The simplified form is:

1045x1510^{\frac{4}{5}} x^{\frac{1}{5}}

This is the expression in terms of fractional exponents.


Would you like further explanation on fractional exponents or the rules used here?

Related Questions:

  1. How do you simplify an expression involving a cube root?
  2. What are the key properties of fractional exponents?
  3. How do you distribute fractional exponents over multiplication?
  4. Can you provide examples of simplifying roots for variables with exponents?
  5. How do fractional exponents relate to logarithms?

Tip:

When simplifying roots, always express them as fractional exponents first—it makes handling the algebra much more straightforward.

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Math Problem Analysis

Mathematical Concepts

Fractional Exponents
Roots and Radicals
Exponent Rules

Formulas

n-th root of a = a^(1/n)
(a * b)^c = a^c * b^c

Theorems

Properties of Exponents
Distributive Property of Exponents over Multiplication

Suitable Grade Level

Grades 8-10