\frac{3}{\sqrt[7]{5^3}}\cdot \ racinaliza

To rationalize the expression 

\[
\frac{3}{\sqrt[7]{5^3}}
\]

we need to remove the seventh root from the denominator. The denominator is currently \( \sqrt[7]{5^3} \), which can be written as \( 5^{3/7} \).

To rationalize, we multiply both the numerator and the denominator by \( 5^{4/7} \), so the exponent in the denominator sums up to a whole number:

\[
\frac{3}{5^{3/7}} \cdot \frac{5^{4/7}}{5^{4/7}} = \frac{3 \cdot 5^{4/7}}{5^{3/7 + 4/7}} = \frac{3 \cdot 5^{4/7}}{5^1} = \frac{3 \cdot 5^{4/7}}{5}
\]

Now, we can simplify:

\[
= \frac{3 \cdot 5^{4/7}}{5} = 3 \cdot 5^{4/7 - 1} = 3 \cdot 5^{-3/7}
\]

Alternatively, we could leave the answer in the form:

\[
\frac{3 \cdot 5^{4/7}}{5}
\]

Would you like further details on any of these steps or the concepts used?

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Here are some related questions that build on this topic:

1. How do you rationalize expressions with cube roots or other non-square roots?
2. What are the general rules for working with fractional exponents?
3. How can you simplify expressions with multiple radical terms in the denominator?
4. Why do we rationalize denominators, and does it change the value of the expression?
5. How would you rationalize a fraction like \( \frac{4}{\sqrt[5]{2}} \)?

**Tip:** When rationalizing a denominator with a root, consider the exponent you need to make the power in the denominator a whole number.

\frac{3}{\sqrt[7]{5^3}} \cdot \text{rationalize}

Learn how to rationalize the expression \frac{3}{\sqrt[7]{5^3}} by removing the seventh root from the denominator. This step-by-step solution explains how to use fractional exponents and properties of exponents, making it suitable for students in Grades 10-12.

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