Simplify each expression. Write answers without negative exponents. Assume all variables represent positive real numbers. (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 like bases: \(a^m \cdot a^n = a^{m+n}\).
3. Add the exponents: \(\frac{7}{3} + \left(-\frac{4}{3}\right) = \frac{7}{3} - \frac{4}{3} = \frac{3}{3}\).
4. Rewrite the expression with the new exponent: \(y^{\frac{3}{3}}\).
5. Simplify the exponent \(\frac{3}{3}\) to 1, so the expression becomes \(y^1\), which is simply \(y\).

Key Concepts

Laws of Exponents

The laws of exponents govern how to simplify expressions involving powers. When multiplying like bases, add the exponents. For example, y^(a) * y^(b) = y^(a+b). This rule is essential for combining terms in the given expression.

Negative Exponents

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

Fractional Exponents

Fractional exponents represent roots and powers simultaneously. For example, y^(m/n) means the nth root of y raised to the mth power. Understanding fractional exponents helps in correctly adding and simplifying the exponents in the expression.