Problem 77 - Elementary and Intermediate Algebra

Question

Simplify each expression. Write the answers without negative exponents. All variables represent positive real numbers. See Example 8. $$ 9^{3 / 7} \cdot 9^{2 / 7} $$

Step-by-Step Solution

Answer

The simplified expression is \( 9^{\frac{5}{7}} \).

Step 1: Understanding the Law of Exponents

When multiplying expressions with the same base, you add the exponents. The general formula is \( a^m \times a^n = a^{m+n} \). In this exercise, the base is 9, and the exponents are \( \frac{3}{7} \) and \( \frac{2}{7} \).

Step 2: Applying the Law of Exponents

Add the exponents \( \frac{3}{7} \) and \( \frac{2}{7} \) together: \[ \frac{3}{7} + \frac{2}{7} = \frac{5}{7} \]. So the expression becomes \( 9^{\frac{5}{7}} \).

Step 3: Writing with Positive Exponents

The expression \( 9^{\frac{5}{7}} \) already has a positive exponent. Since there are no negative exponents present, the simplified form is \( 9^{\frac{5}{7}} \).

Key Concepts

Multiplying Exponents

When faced with an expression involving multiplying exponents, the law of exponents provides a handy way to simplify the process. According to this rule, when you multiply exponents with the same base, you simply add their exponents together.
This is based on the general formula: \(a^m \times a^n = a^{m+n}\).
For example, in the expression \(9^{3/7} \cdot 9^{2/7}\), the base is 9 and applies the simple operation of addition to the exponents:

• Add \(\frac{3}{7} \) to \(\frac{2}{7} \).
• This simplifies the expression to \(9^{(\frac{3}{7} + \frac{2}{7})} \) or \(9^{5/7}\).

You'll notice how the process involves straightforward arithmetic with fractions, which makes it easy to handle even complex expressions.

Expressions with Same Base

In mathematical expressions like the one we're dealing with, identifying terms with the same base is crucial.
The base of an expression is the number used a repeated multiplication, denoted by an exponent.
For expressions that have the same base, you can apply certain rules to simplify calculations.
For instance, in this example, both terms have 9 as their base.

• Being consistent allows us to readily add the exponents together under the same general rule.
• It also means that our final base in the simplified expression remains unchanged.

This principle simplifies computation because no additional multiplication is needed beyond adding exponents. Simple recognition of similar bases streamlines operations drastically.

Positive Exponents

Exponents are key to understanding how numbers behave in exponential expressions and calculations.
In mathematics, a positive exponent tells us how many times to multiply the base by itself.
It's important to ensure expressions are simplified in such a way that exponents remain positive.
It's often a requirement in exercises, since it offers a standard format for showing final results.

• A positive exponent maintains the size of the number, making calculations straightforward to interpret.
• For this exercise, we initially handled positive fractional exponents and ended with \(9^{5/7}\), which is already positive.

Managing positive exponents allows for consistency and clarity when presenting and understanding results.