Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. 4r^-3/6r^-6

Verified step by step guidance

1. Start by writing the expression clearly: \(\frac{4r^{-3}}{6r^{-6}}\).
2. Separate the coefficients and the variable parts: \(\frac{4}{6} \times \frac{r^{-3}}{r^{-6}}\).
3. Simplify the coefficients by dividing 4 by 6: \(\frac{4}{6} = \frac{2}{3}\).
4. Use the quotient rule for exponents on the variable part: \(\frac{r^{-3}}{r^{-6}} = r^{-3 - (-6)} = r^{3}\).
5. Combine the simplified coefficient and variable parts: \(\frac{2}{3} \times r^{3}\), which is the expression without negative exponents.

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers, such as multiplying, dividing, and raising powers to powers. For division, subtract the exponents of like bases (a^m / a^n = a^(m-n)). Understanding these rules is essential to simplify expressions with negative exponents.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the positive exponent (a^(-n) = 1/a^n). To write answers without negative exponents, rewrite terms with negative exponents as fractions with positive exponents in the numerator or denominator.

Simplifying Rational Expressions

Simplifying rational expressions involves reducing fractions by canceling common factors in the numerator and denominator. When variables with exponents appear, apply exponent rules carefully to combine and reduce terms, ensuring the final expression is in simplest form.