Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. (5x)^-2(5x^3)^-3/(5^-2x^-3)^-3
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Start by applying the property of exponents to each term in the expression: , , \text{and} \ ((5^{-2}x^{-3})^{-3}\).
Simplify each term: , , \text{and} \ ((5^{-2}x^{-3})^{-3} = 5^6x^9\).
Combine the terms in the numerator: .
Now, divide the combined numerator by the simplified denominator: .
Apply the quotient rule for exponents to simplify: .
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Negative Exponents
Negative exponents indicate the reciprocal of the base raised to the absolute value of the exponent. For example, a^-n = 1/a^n. This concept is crucial for simplifying expressions with negative exponents, as it allows us to rewrite them in a more manageable form, eliminating the negatives in the process.
Exponent rules, such as the product of powers, power of a power, and quotient of powers, govern how to manipulate expressions involving exponents. For instance, (a^m)(a^n) = a^(m+n) and (a^m)/(a^n) = a^(m-n). Understanding these rules is essential for simplifying complex expressions accurately.
Simplifying rational expressions involves reducing fractions to their simplest form by canceling common factors in the numerator and denominator. This process often requires applying the rules of exponents and ensuring that all variables are treated as nonzero, which is critical for avoiding undefined expressions.