what should be multiplied to nth root  100 ^ 5 so that it will become a rational number

Given the expression \( 100^{\frac{5}{n}} \), we want to find a value that, when multiplied by this expression, results in a rational number.

### Step-by-Step Breakdown

1. **Understanding the Problem**:
   - The expression \( 100^{\frac{5}{n}} \) is the \( n \)th root of \( 100^5 \).
   - A number in the form \( a^{\frac{m}{n}} \) is rational if \( \frac{m}{n} \) simplifies to an integer.
   - Thus, we need to multiply \( 100^{\frac{5}{n}} \) by a number that makes the exponent an integer.

2. **Simplifying the Expression**:
   - \( 100 = 10^2 \), so \( 100^5 = (10^2)^5 = 10^{10} \).
   - This means \( 100^{\frac{5}{n}} = 10^{\frac{10}{n}} \).

3. **Finding the Multiplier**:
   - To make \( 10^{\frac{10}{n}} \) a rational number, the exponent \( \frac{10}{n} \) must be an integer.
   - We need to multiply \( 10^{\frac{10}{n}} \) by some \( 10^k \) such that the total exponent becomes an integer:
     \[
     10^{\frac{10}{n}} \times 10^k = 10^{\left(\frac{10}{n} + k\right)}.
     \]
   - For \( \frac{10}{n} + k \) to be an integer, \( \frac{10}{n} \) must either already be an integer or \( k \) must be chosen to make up for any fractional part.

4. **Determining the Required Multiplier**:
   - Let’s rewrite the expression: multiply \( 100^{\frac{5}{n}} = 10^{\frac{10}{n}} \) by \( 100^{\frac{m}{n}} = 10^{\frac{2m}{n}} \).
   - Find \( m

what should be multiplied to nth root 100 ^ 5 so that it will become a rational number

Explore how to multiply the nth root of 100^5 to form a rational number. This solution breaks down the steps using exponent rules, focusing on how to simplify the expression 100^(5/n). Suitable for students in Grades 9-12.

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