Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. (5x)^-2(5x³)^-3/(5^-2x^-3)^-3

Verified step by step guidance

1. Rewrite the expression clearly as a fraction: \(\frac{(5x)^{-2} (5x^{3})^{-3}}{(5^{-2} x^{-3})^{-3}}\).
2. Apply the power of a product rule to each term: \((ab)^m = a^m b^m\). So, rewrite \((5x)^{-2}\) as \$5^{-2} x^{-2}\( and \)(5x^{3})^{-3}\( as \)5^{-3} (x^{3})^{-3} = 5^{-3} x^{-9}$.
3. Simplify the denominator by applying the power of a power rule: \((a^m)^n = a^{mn}\). So, \((5^{-2} x^{-3})^{-3} = 5^{(-2)(-3)} x^{(-3)(-3)} = 5^{6} x^{9}\).
4. Combine all terms in numerator and denominator: numerator becomes \$5^{-2} x^{-2} \times 5^{-3} x^{-9} = 5^{-5} x^{-11}\(, denominator is \)5^{6} x^{9}$.
5. Divide the numerator by the denominator by subtracting exponents of like bases: \$5^{-5 - 6} x^{-11 - 9} = 5^{-11} x^{-20}\(. Then rewrite without negative exponents by using \)a^{-m} = \frac{1}{a^{m}}$.

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers, such as multiplying powers with the same base by adding exponents, raising a power to another power by multiplying exponents, and dividing powers by subtracting exponents. Understanding these rules is essential for simplifying expressions like the given problem.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the corresponding positive exponent. For example, x^(-n) equals 1/x^n. Converting negative exponents to positive ones is crucial for writing the final answer without negative exponents, as requested in the problem.

Simplifying Algebraic Fractions

Simplifying algebraic fractions involves applying exponent rules carefully to both numerator and denominator, then reducing the expression by canceling common factors. This process helps in rewriting complex expressions into simpler forms, especially when variables and exponents are involved.