Problem 32 - A Graphical Approach to College Algebra

Question

Perform the indicated operations. Write your answers with only positive exponents. Assume that all variables represent positive real numbers. $$\left(x^{4 / 5}\right)\left(x^{2 / 5}\right)$$

Step-by-Step Solution

Answer

The result is \(x^{6/5}\).

Step 1: Identify the Operation

The problem involves multiplying two powers with the same base \(x\). According to the laws of exponents, to multiply like bases, you add the exponents.

Step 2: Sum the Exponents

Add the exponents of \(x\): \(\frac{4}{5} + \frac{2}{5} = \frac{6}{5}\). This step is done by ensuring the fractions have a common denominator, which they do.

Step 3: Write the Result

The expression becomes \(x^{6/5}\) after adding the exponents.

Key Concepts

Laws of Exponents

The laws of exponents are simple yet powerful rules that help us manipulate and simplify expressions involving powers. When dealing with exponents, there are several key rules to remember. One crucial rule is that when you multiply powers with the same base, you add their exponents. This is known as the Product of Powers Property.
For example, if you have two exponents with the same base, such as \(x^a\) and \(x^b\), you can multiply them by simply adding the exponents:

• \(x^a \cdot x^b = x^{a+b}\)

This rule makes it easier to work with complex expressions because it reduces the number of steps needed to find the solution. Remembering to add the exponents when you multiply powers with like bases is pivotal in simplifying mathematical expressions effectively.

Multiplying Powers

Multiplying powers is straightforward if you stick to the rules. The core principle is that when you have the same base for two or more powers, you apply the laws of exponents to add their exponents together. Consider the expression \(\left(x^{\frac{4}{5}}\right) \left(x^{\frac{2}{5}}\right)\):

• Both powers have the base \(x\), allowing us to add their exponents: \(\frac{4}{5} + \frac{2}{5}\).
• The resultant expression is \(x^{\frac{6}{5}}\), since \(\frac{4}{5} + \frac{2}{5} = \frac{6}{5}\).

By following this approach, you simplify the original equation into a simpler form, helping you solve math problems faster and with fewer mistakes.

Positive Exponents

Positive exponents indicate how many times a number, the base, is multiplied by itself. They are an essential concept in math and physics, providing a way to express repeated multiplication compactly. When working with exponents, it's often important to present your final answer using only positive exponents, as this is the standard form.

• For instance, \(x^{-n} = \frac{1}{x^n}\). By converting negative exponents to positive ones using this rule, your answers remain clear and standard.
• In expressions such as \(x^{6/5}\), the exponent \(6/5\) is positive, indicating the base \(x\) raised to the power of \(6/5\).

Employing positive exponents in your solutions is crucial for clarity and consistency, ensuring that all students and mathematicians can follow and understand your results.