Simplifying algebraic expressions

Exo 1

Simplify the following expressions with positive exponents.

\((5x^{-7}y^3)^{-5}\)

Solution:

Step 1: Apply the exponent to each term inside the parentheses
Use the property \( (ab)^n = a^n \cdot b^n \) to distribute the exponent \(-5\) to each factor:
\[
(5x^{-7}y^3)^{-5} = 5^{-5} \cdot (x^{-7})^{-5} \cdot (y^3)^{-5}
\]

Step 2: Simplify each term

\( 5^{-5} = \frac{1}{5^5} \)
\( (x^{-7})^{-5} = x^{(-7) \cdot (-5)} = x^{35} \)
\( (y^3)^{-5} = y^{3 \cdot (-5)} = y^{-15} \)

Step 3: Combine the results
Now, putting everything together:
\[
5^{-5} \cdot x^{35} \cdot y^{-15} = \frac{1}{5^5} \cdot x^{35} \cdot \frac{1}{y^{15}}
\]

Final Answer:
\[
\frac{x^{35}}{5^5 y^{15}}
\]
Thus, the simplified expression with positive exponents is:

\[
\frac{x^{35}}{5^5 y^{15}}
\]