Question

Simplify each expression. Express your answer so that only positive exponents occur. Assume that the variables are positive. $$x^{2 / 3} x^{1 / 2} x^{-1 / 4}$$

Short answer

The simplified expression is \(x^{11/12}\).

Step-by-step solution

Apply the Product of Powers Property

When multiplying expressions with the same base, add the exponents. In this case, combine the exponents of the base \(x\): \(x^{2/3} x^{1/2} x^{-1/4} = x^{(2/3 + 1/2 + (-1/4))}\).

Convert Exponents to a Common Denominator

Convert the fractions \(2/3, 1/2,\) and \(-1/4\) to have a common denominator. The least common denominator of 3, 2, and 4 is 12. \[ \frac{2}{3} = \frac{8}{12}, \frac{1}{2} = \frac{6}{12}, \frac{-1}{4} = \frac{-3}{12} \]

Add the Exponents

Now add the adjusted exponents: \[ \frac{8}{12} + \frac{6}{12} + \-\frac{3}{12} = \frac{11}{12}. \] This results in \(x^{11/12}\).

Confirm Positive Exponents

Ensure that all exponents are positive, which they are in this case. The simplified expression is \(x^{11/12}\).

Key concepts

Positive Exponents

In algebra, an exponent tells us how many times to multiply the base by itself. A positive exponent means we are multiplying the base several times. For instance, in the expression \(x^3\), the exponent 3 indicates that we multiply x by itself three times (\(x \cdot x \cdot x\)).

When simplifying expressions, it's crucial to ensure that all final exponents are positive. Negative exponents indicate reciprocal operations. For example, \(x^{-3} = \frac{1}{x^3}\). In our exercise, the goal is to have only positive exponents in the final simplified expression.

This is achieved by correctly applying algebraic properties like the Product of Powers Property and ensuring no negative exponents remain in the result. Ensuring positive exponents helps simplify and standardize expressions for easier interpretation and further calculations.

Product of Powers Property

When multiplying expressions with the same base, you can simplify by adding their exponents. This is called the Product of Powers Property. Here's how it works:

Given an expression like \x^a \cdot x^b\, you apply the property to combine the exponents: \x^a \cdot x^b = x^{(a+b)}\.

In our exercise, we used this property to simplify \x^{2/3} \cdot x^{1/2} \cdot x^{-1/4}\ by combining the exponents of the base x. The new exponent becomes \(2/3 + 1/2 + (-1/4)\).

Step by step:

• Recognize all components have the same base (x).
• Use the Product of Powers Property: \(x^{2/3} \cdot x^{1/2} \cdot x^{-1/4} = x^{(2/3 + 1/2 + (-1/4))}\).

This method helps to keep things straightforward and ensures expressions are correctly simplified when working with power-based terms.

Common Denominator

Adding fractions directly requires a common denominator. A common denominator is a shared multiple of the denominators in each fraction.

For the exercise, we need to add exponents \(\frac{2}{3}\), \(\frac{1}{2}\), and \(\frac{-1}{4}\). Their denominators are 3, 2, and 4, respectively.

To make the addition possible, we find the least common denominator (LCD). The LCD of 3, 2, and 4 is 12. Next, convert each fraction:

• \(\frac{2}{3} = \frac{8}{12}\)
• \(\frac{1}{2} = \frac{6}{12}\)
• \(\frac{-1}{4} = \frac{-3}{12}\)

Now, add the fractions: \[ \frac{8}{12} + \frac{6}{12} + \frac{-3}{12} = \frac{11}{12}\. \]

Hence, the simplified exponent for base x is \frac{11}{12}\, ensuring the expression \(x^{11/12}\) operates purely with positive exponents.