Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. y⁸/y¹²

Verified step by step guidance

1. Identify the expression to simplify: \(\frac{y^8}{y^{12}}\).
2. Recall the quotient rule for exponents: \(\frac{a^m}{a^n} = a^{m-n}\), where \(a \neq 0\).
3. Apply the quotient rule to the expression: \(y^{8-12} = y^{-4}\).
4. Rewrite the expression to eliminate the negative exponent by using the rule \(a^{-m} = \frac{1}{a^m}\): \(y^{-4} = \frac{1}{y^4}\).
5. Write the final simplified expression as \(\frac{1}{y^4}\), which has no negative exponents.

Key Concepts

Laws of Exponents

The laws of exponents provide rules for simplifying 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 is essential for simplifying expressions like y^8 / y^12.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the positive exponent, such as a^(-n) = 1 / a^n. Since the problem requires answers without negative exponents, any negative exponent must be rewritten as a positive exponent in the denominator or numerator.

Assumption of Nonzero Variables

Assuming variables represent nonzero real numbers ensures that division by zero does not occur and that exponent rules apply correctly. This assumption allows simplification without concern for undefined expressions or zero denominators.