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

You are asking for what should be multiplied to \( \sqrt[n]{100^5} \) so that the result becomes a rational number.

### Given Expression: 
\[
\sqrt[n]{100^5} = (100^5)^{1/n}
\]

To make this expression a rational number, we need the exponent of the base (100) to be an integer after applying the root. This means we need to adjust the power of \( 100^5 \) so that when divided by \( n \), it results in an integer.

### Required Condition:
For the expression to become rational, the total exponent of 100 must be divisible by \( n \). Currently, we have:
\[
100^{5/n}
\]
We want the power of 100 to become an integer, so we need to multiply the original expression by something that will result in an integer exponent.

To do this, multiply \( \sqrt[n]{100^5} \) by \( \sqrt[n]{100^{n-5}} \), so that the total power of 100 becomes \( 100^n \), which simplifies to \( 100 \), a rational number.

### Multiplication Needed:
You should multiply \( \sqrt[n]{100^5} \) by \( \sqrt[n]{100^{n-5}} \) to make the result a rational number.

Thus, the expression becomes:
\[
\sqrt[n]{100^5} \times \sqrt[n]{100^{n-5}} = \sqrt[n]{100^n} = 100
\]

### Answer:
The required multiplier is \( \sqrt[n]{100^{n-5}} \).

Would you like more details or clarification? 

Here are 5 related questions to deepen your understanding:
1. What is the difference between rational and irrational numbers?
2. How can you simplify the nth root of a number in general?
3. Why does multiplying powers of the same base involve adding exponents?
4. What happens when you multiply two roots with the same base?
5. How would this approach change if we were working with an irrational base?

**Tip**: When working with roots and exponents, remember that powers can be added when multiplying expressions with the same base.

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

Find out what should be multiplied to the nth root of 100^5 to make the result a rational number. This problem explores key concepts such as roots, exponents, and rational numbers, providing a detailed solution. Suitable for students in Grades 10-12.

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