Problem 49 - Algebra and Trigonometry

Question

\(49-68=\) Simplify the expression and eliminate any negative exponent(s). Assume that all letters denote positive numbers. $$ x^{3 / 4} x^{5 / 4} $$

Step-by-Step Solution

Answer

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

Step 1: Combine the Exponents

When multiplying expressions with the same base, add the exponents. Here, you are multiplying two expressions with base \(x\). The exponents are \(\frac{3}{4}\) and \(\frac{5}{4}\). Thus, you can add these exponents together: \[x^{\frac{3}{4} + \frac{5}{4}}\]

Step 2: Simplify the Exponent Addition

Add the exponents: \[\frac{3}{4} + \frac{5}{4} = \frac{8}{4} = 2\]Now, you replace the expression in the exponent with its simplified form. Your expression now becomes: \[x^2\]

Step 3: Rewrite the Expression

By simplifying \(x^{2}\), we find that the expression is already in its simplest form. There are no negative exponents present, so there's no need for further modification.

Key Concepts

Multiplying Expressions

When it comes to multiplying expressions, especially those involving exponents, it’s essential to recognize the base of the expressions. A base is the number that is being multiplied by itself, as indicated by the exponent. In the given exercise, both expressions share the same base: \(x\). This common base allows us to use the properties of exponents effectively.
Multiply the expressions by first identifying the bases. If the bases are the same, you can proceed to the next step. Always look for opportunities to simplify expressions by using the base-exponent property.
In addition, it is helpful to know that multiplying expressions with exponents is less about the numbers themselves and more about the rules governing these operations. Recognizing whether bases are the same or different is the first step in applying these rules correctly.

Exponent Addition

Exponent addition is integral in simplifying expressions with the same base. This process involves adding the exponents when the bases are the same. For example, in the exercise, we have \(x^{3/4}\) and \(x^{5/4}\) with a common base of \(x\).
Here’s how it works:

• Identify the expressions with the same base, as with \(x\) in our example.
• Add the exponents together: \(\frac{3}{4} + \frac{5}{4}\).

This step leads us to add the fractions. After performing the addition \(\frac{3}{4} + \frac{5}{4} = \frac{8}{4}\), which simplifies to 2. Hence, the expression now transforms into \(x^2\).
Remember, exponent addition is simple when fractions are involved. Just ensure the fractions have the same denominator before adding them. This keeps the process straightforward and manageable.

Simplifying Exponents

Finally, the goal of any problem involving exponents is to simplify as much as possible. Simplifying exponents means rewriting them in a form that is easy to understand and compute. After completing the exponent addition in our case, you arrive at \(x^2\).
Here's what to do next:

• Check for negative exponents or any possibility of further simplification.
• Ensure that the new expression is as simplified as it can be.

In this specific exercise, \(x^2\) does not involve negative exponents. Therefore, it remains in its simplest form.
By understanding how to simplify correctly, we ensure clarity and accuracy in our results. Always observe any rules or properties of exponents that might help further in simplifying beyond the initial steps.