Simplify each expression. Write answers without negative exponents. Assume all variables represent nonzero real numbers. 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 by subtracting: \$7 - 10 = -3\(, so the expression becomes \)r^{-3}$.
5. Rewrite the expression without negative exponents by using the rule \(a^{-m} = \frac{1}{a^{m}}\), resulting in \(\frac{1}{r^{3}}\).

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers. For division, subtract the exponent in the denominator from the exponent in the numerator when the bases are the same, e.g., r^7 / r^10 = r^(7-10) = r^(-3).

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the positive exponent. For example, r^(-3) equals 1 / r^3. To write answers without negative exponents, rewrite expressions using positive exponents in the denominator.

Simplification of Algebraic Expressions

Simplifying expressions involves rewriting them in a simpler or more standard form. This includes applying exponent rules correctly and expressing the final answer without negative exponents, assuming variables are nonzero to avoid division by zero.