Simplify each expression. Write answers without negative exponents. Assume all variables represent nonzero real numbers. 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: when dividing like bases, subtract the exponents, so \(\frac{a^{m}}{a^{n}} = a^{m-n}\).
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 \(a^{-m} = \frac{1}{a^{m}}\), so \(y^{-4} = \frac{1}{y^{4}}\).
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 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., y^8 / y^12 = y^(8-12) = y^(-4).

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the positive exponent. For example, y^(-4) equals 1/y^4. Writing answers without negative exponents means converting expressions like y^(-4) into 1/y^4.

Assumption of Nonzero Variables

Assuming variables represent nonzero real numbers ensures that division by zero does not occur and that expressions with negative exponents are valid. This assumption allows simplification without restrictions on the domain.