Simplify each expression. Write answers without negative exponents. Assume all vari-ables represent positive real numbers. See Examples 8 and 9. (y^7/3)(y^-4/3)

Verified step by step guidance

1. Identify the expression to simplify: \(\left(y^{\frac{7}{3}}\right) \left(y^{-\frac{4}{3}}\right)\).
2. Recall the property of exponents for multiplying powers with the same base: \(a^m \cdot a^n = a^{m+n}\).
3. Add the exponents: \(\frac{7}{3} + \left(-\frac{4}{3}\right) = \frac{7}{3} - \frac{4}{3}\).
4. Simplify the sum of the exponents: \(\frac{7}{3} - \frac{4}{3} = \frac{3}{3} = 1\).
5. Rewrite the expression with the simplified exponent: \(y^1\), which is simply \(y\). This expression has no negative exponents.

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers. When multiplying like bases, add the exponents (e.g., a^m * a^n = a^(m+n)). This rule is essential for combining terms like (y^(7/3)) and (y^(-4/3)) into a single power.

Negative Exponents

A negative exponent indicates the reciprocal of the base raised to the positive exponent (a^(-n) = 1/a^n). To write answers without negative exponents, rewrite terms with negative powers as fractions with positive exponents in the denominator.

Simplifying Expressions with Rational Exponents

Rational exponents represent roots and powers simultaneously (e.g., y^(m/n) = n-th root of y^m). Understanding how to manipulate and combine rational exponents is crucial for simplifying expressions like (y^(7/3)) * (y^(-4/3)) accurately.