Problem 50 - Algebra and Trigonometry

Question

\(49-68=\) Simplify the expression and eliminate any negative exponent(s). Assume that all letters denote positive numbers. $$ y^{2 / 3} y^{4 / 3} $$

Step-by-Step Solution

Answer

The simplified expression is \( y^2 \).

Step 1: Simplify the Fractional Exponents

The given expression is \( y^{2/3} \times y^{4/3} \). When multiplying expressions with the same base, you add the exponents. Therefore, we add \( \frac{2}{3} \) and \( \frac{4}{3} \).

Step 2: Add the Exponents

Add the fractional exponents: \( \frac{2}{3} + \frac{4}{3} = \frac{2 + 4}{3} = \frac{6}{3} \).

Step 3: Simplify the Resulting Exponent

Simplify \( \frac{6}{3} \): \( \frac{6}{3} = 2 \). Hence, the expression simplifies to \( y^2 \).

Key Concepts

Fractional Exponents

Fractional exponents, also known as rational exponents, are expressions where the exponent is a fraction instead of a whole number. The numerator of the fraction represents the power, while the denominator indicates the root. For example, the expression \( x^{1/2} \) translates to the square root of \( x \). A fractional exponent like \( y^{2/3} \) means you have a cube root (due to the 3 in the denominator) of \( y^2 \). When working with fractional exponents, it's important to understand this dual interpretation of roots and powers, which provides a flexible way to manipulate and simplify expressions with roots and powers.

Fractional exponents can make complex expressions more manageable and allow us to apply the operations of exponentiation and root extraction systematically.

Simplifying Expressions

Simplifying mathematical expressions involves reducing them to their simplest form while keeping the underlying value intact. For example, when given the expression \( y^{2/3} \times y^{4/3} \), you can utilize rules that make handling fractional exponents easier.

An essential step in simplifying is recognizing expressions with the same base. When multiplying such expressions, like in our example, you add the exponents together. So, \( y^{2/3} \times y^{4/3} \) can be simplified by adding the exponents: \( \frac{2}{3} + \frac{4}{3} = \frac{6}{3} \). Once you simplify the resulting fraction \( \frac{6}{3} \), you get the integer 2, leading to the simplified expression \( y^2 \).

The goal of simplification is to make expressions easier to work with, which is critical in solving equations or evaluating expressions at specific points.

Properties of Exponents

The properties of exponents are fundamental rules that govern how to manipulate expressions involving powers. These rules are especially helpful in simplifying expressions and solving equations with exponents. Here are some key properties:

• **Product of Powers:** When multiplying two exponents with the same base, add their exponents. This property is shown in the expression \( y^{2/3} \times y^{4/3} = y^{(2/3 + 4/3)} \).
• **Quotient of Powers:** When dividing, subtract the exponents if the bases are the same.
• **Power of a Power:** To raise a power to another power, multiply the exponents.
• **Power of a Product:** Distribute the exponent to each base in a product.

These properties make working with exponents much more straightforward and allow for systematic simplification and manipulation of expressions. Understanding these principles is crucial in not just simplifying expressions, but also in solving various mathematical problems involving exponents.