Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. r⁷/r¹⁰

Verified step by step guidance

1. Identify the expression to simplify: \(\frac{r^7}{r^{10}}\).
2. Recall the quotient rule for exponents, which states that \(\frac{a^m}{a^n} = a^{m-n}\) when \(a \neq 0\).
3. Apply the quotient rule to the expression: \(\frac{r^7}{r^{10}} = r^{7-10}\).
4. Simplify the exponent: \(r^{7-10} = r^{-3}\).
5. Rewrite the expression without negative exponents by using the rule \(a^{-m} = \frac{1}{a^m}\), so \(r^{-3} = \frac{1}{r^3}\).

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers. For division, subtract the exponent of the denominator from the exponent of the numerator when the bases are the same, e.g., a^m / a^n = a^(m-n). This rule is essential for simplifying expressions like r^7 / r^10.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the positive exponent, such as a^(-n) = 1 / a^n. When simplifying, answers should be rewritten to avoid negative exponents by expressing them as fractions with positive exponents.

Assumption of Nonzero Variables

Assuming variables represent nonzero real numbers ensures that division by zero does not occur. This assumption allows the use of exponent rules safely, as zero bases with negative exponents are undefined.