Problem 77 - 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 expression simplifies to \(9^{5/7}\).

Step 1: Understand the Rule of Exponents

When multiplying powers with the same base, add the exponents. This can be expressed by the formula: \(a^m \cdot a^n = a^{m+n}\).

Step 2: Identify the Base and Exponents

In the given expression \(9^{3/7} \cdot 9^{2/7}\), both terms have the base of 9. The exponents are \(\frac{3}{7}\) and \(\frac{2}{7}\).

Step 3: Add the Exponents

Apply the rule from Step 1: add the exponents \(\frac{3}{7} + \frac{2}{7}\). This gives \(\frac{5}{7}\).

Step 4: Simplify the Expression

The expression \(9^{3/7} \cdot 9^{2/7}\) simplifies to \(9^{5/7}\) by applying the rule and adding the exponents.

Key Concepts

Simplifying Expressions

When you simplify expressions involving exponents, the goal is to make them more manageable and easier to understand. You do that by applying exponent rules that help you combine or break down expressions.
One common rule is that when multiplying powers with the same base, you add the exponents. For instance, if you have \(a^m \cdot a^n\), you can simplify this to \(a^{m+n}\). In our example, it means you would add the fractional exponents of the same base number.

• Identify the base number shared by terms.
• Add the exponents if the base is the same.
• Rewrite the expression with the new single exponent.

These steps ensure that the expression is expressed in the simplest form, which is crucial for further operations or evaluations.

Fractional Exponents

Fractional exponents might seem tricky at first, but they're just another way to represent roots and powers. A fractional exponent like \(a^{m/n}\) indicates a root as well as a power. Here’s how to interpret it:

• The number \(m\) is the power, and \(n\) is the root.
• The expression \(a^{m/n}\) is the same as \(\sqrt[n]{a^m}\).

In the example \(9^{5/7}\), this expression can be interpreted as the 7th root of \(9^5\). This way of writing helps especially when dealing with more complex roots and powers, showing exactly what operation is being performed.

Positive Real Numbers

When working with exponents, particularly fractional ones, it is essential to consider the positive real numbers. This is because fractional exponents often imply taking roots, and roots of negative numbers can lead to complex or unintelligible results when not dealing with real numbers.
By defining variables to represent positive real numbers, you can ensure the results remain real and easily understandable. This is crucial because negative bases could turn fractional roots into imaginary numbers, which may not be desired in certain contexts.

• Stick to positive bases with fractional exponents for real results.
• Understand the implications of roots to avoid complex solutions.

Overall, when simplifying expressions, using positive real numbers aligns with the intention to keep the expressions real and applicable.