Perform the indicated operation, and write each answer in lowest terms. y³/8 ÷ y/4

Verified step by step guidance

1. Rewrite the division problem as a multiplication by the reciprocal. That is, change \(\frac{y^{3}}{8} \div \frac{y}{4}\) to \(\frac{y^{3}}{8} \times \frac{4}{y}\).
2. Multiply the numerators together and the denominators together: \(\frac{y^{3} \times 4}{8 \times y}\).
3. Simplify the coefficients (numbers) by dividing 4 and 8 by their greatest common divisor: \(\frac{4}{8} = \frac{1}{2}\), so the expression becomes \(\frac{y^{3} \times 1}{2 \times y}\).
4. Simplify the variables using the laws of exponents. Since you have \(\frac{y^{3}}{y}\), subtract the exponents: \(y^{3-1} = y^{2}\), so the expression is now \(\frac{y^{2}}{2}\).
5. Write the final simplified expression as \(\frac{y^{2}}{2}\), which is the answer in lowest terms.

Key Concepts

Division of Fractions

Dividing fractions involves multiplying the first fraction by the reciprocal of the second. For example, to divide a/b by c/d, multiply a/b by d/c. This method simplifies the division process and is essential for solving fraction division problems.

Laws of Exponents

When dividing expressions with the same base, subtract the exponents: a^m ÷ a^n = a^(m-n). This rule helps simplify expressions involving variables raised to powers, such as y^3 ÷ y^1 = y^(3-1) = y^2.

Simplifying Fractions

After performing operations, fractions should be simplified to their lowest terms by dividing numerator and denominator by their greatest common divisor. Simplifying ensures the answer is expressed in the simplest and most understandable form.