Simplify each expression. Write answers without negative exponents. Assume all variables represent nonzero real numbers. 4r^-3/6r^-6

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Start with the given expression: \(\frac{4r^{-3}}{6r^{-6}}\).

Rewrite the expression by separating the coefficients and the variable parts: \(\frac{4}{6} \times \frac{r^{-3}}{r^{-6}}\).

Simplify the coefficient fraction \(\frac{4}{6}\) by dividing numerator and denominator by their greatest common divisor.

Apply the quotient rule for exponents to the variable part: \(\frac{r^{a}}{r^{b}} = r^{a - b}\), so \(\frac{r^{-3}}{r^{-6}} = r^{-3 - (-6)}\).

Simplify the exponent expression and rewrite the result without negative exponents by using the rule \(r^{-n} = \frac{1}{r^{n}}\) if needed.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers. Key rules include dividing powers with the same base by subtracting exponents and rewriting negative exponents as positive by taking reciprocals. For example, a^m / a^n = a^(m-n), and a^(-k) = 1/a^k.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions by factoring and canceling common terms in numerator and denominator. When variables with exponents appear, apply exponent rules to combine or reduce powers before simplifying the fraction.

Eliminating Negative Exponents

Negative exponents indicate reciprocals and must be rewritten as positive exponents for final answers. For example, x^(-n) is rewritten as 1/x^n. This step ensures expressions are presented in standard form without negative powers.

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