Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent nonzero real numbers. See Examples 5 and 6. x¹²/x⁸

Verified step by step guidance

1. Identify the expression to simplify: \(\frac{x^{12}}{x^{8}}\).
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: \(\frac{x^{12}}{x^{8}} = x^{12-8}\).
4. Simplify the exponent by subtracting: \(x^{12-8} = x^{4}\).
5. Since the exponent is positive, the expression is already written without negative exponents, so the simplified form is \(x^{4}\).

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., x^a / x^b = x^(a-b). This rule is essential for simplifying expressions like x^12 / x^8.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the positive exponent, such as x^(-n) = 1/x^n. Since the problem requires answers without negative exponents, it is important to rewrite any negative exponents as positive by moving factors between numerator and denominator.

Assumption of Nonzero Variables

Assuming variables represent nonzero real numbers ensures that division by zero does not occur, which is undefined. This assumption allows the use of exponent rules safely, especially when simplifying expressions involving variables in denominators.