Problem 33 - 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. $$(1+n)^{1 / 2}(1+n)^{3 / 4}$$

Step-by-Step Solution

Answer

\((1+n)^{5/4}\)

Step 1: Understand the Problem

We need to simplify the expression \((1+n)^{1/2} (1+n)^{3/4}\) by performing the indicated operations and expressing the result with positive exponents.

Step 2: Apply the Product of Powers Property

The Product of Powers Property states that when you multiply two powers with the same base, you add their exponents. The expression is \((1+n)^{1/2} (1+n)^{3/4}\), so we add the exponents: \(\frac{1}{2} + \frac{3}{4}\).

Step 3: Find a Common Denominator

To add \(\frac{1}{2}\) and \(\frac{3}{4}\), we need a common denominator. The least common denominator is 4. Convert \(\frac{1}{2}\) to \(\frac{2}{4}\) so both fractions have the same denominator.

Step 4: Add the Exponents

Now add the exponents: \(\frac{2}{4} + \frac{3}{4} = \frac{5}{4}\).

Step 5: Write the Result with Positive Exponents

The simplified expression is \((1+n)^{5/4}\), which is already expressed with positive exponents.

Key Concepts

Product of Powers

Exponentiation uses the property known as the "product of powers" to make multiplying expressions easier. When you have two or more powers with the same base, this rule states you can add their exponents to simplify the expression. Consider the example expression \( (1+n)^{1/2} (1+n)^{3/4} \). Both parts share the base \( 1+n \). According to the product of powers property, we add their exponents to combine them:

• Add \( \frac{1}{2} \) and \( \frac{3}{4} \).

This turns the expression into a single power with a sum of the exponents. The result is a simplified exponent: \( (1+n)^{5/4} \). Using this property is handy for making expressions more manageable to work with.
Remember, this only applies when powers share the same base.

Common Denominator

To add fractions, it's crucial to have a common denominator. This means converting fractions to have the same bottom number so they can be directly added. In our example:

• Fractions are \( \frac{1}{2} \) and \( \frac{3}{4} \).

The least common denominator, or smallest number both denominators go into, is 4. So, convert \( \frac{1}{2} \) to \( \frac{2}{4} \), preserving the value but allowing us to add the fractions easily:
\( \frac{2}{4} + \frac{3}{4} = \frac{5}{4} \). Having a common denominator is crucial for correct and straightforward addition of fractions. When dealing with exponents, adding is only possible if you ensure a common denominator.

Positive Exponents

In mathematics, it's often standard to write expressions using positive exponents to emphasize straightforward and simplified representation.
Positive exponents indicate the number of times you multiply a number by itself. Here, the problem asks us to provide the final answer with positive exponents. For example, \( (1+n)^{5/4} \) is already in the desired form.
A positive exponent like \( 5/4 \) indicates that \( 1+n \) is a base multiplied by itself in increasing fractions of one. This is much more practical and readable than dealing with negative exponents, which suggest division or reciprocal values. Always aim to simplify your work by expressing final answers in terms of positive exponents.