Problem 19 - College Algebra with Corequisite Support

Question

For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents. \(7^{-6} \times 7^{-3}\)

Step-by-Step Solution

Answer

The expression simplifies to \(\frac{1}{7^9}\).

Step 1: Identify the Bases

Look at the given expression, \(7^{-6} \times 7^{-3}\). Both terms share the same base, which is 7.

Step 2: Apply the Product of Powers Property

The product of powers property states that when multiplying like bases, you add the exponents. For the expression \(7^{-6} \times 7^{-3}\), add the exponents \(-6\) and \(-3\).

Step 3: Calculate the Sum of the Exponents

Add the exponents: \(-6 + (-3) = -9\).

Step 4: Write the Expression with a Single Base

Combine the base with the new exponent, resulting in \(7^{-9}\).

Step 5: Convert to Positive Exponents

Since the problem requests positive exponents, rewrite \(7^{-9}\) using a positive exponent as \(\frac{1}{7^9}\).

Key Concepts

Product of Powers Property

The product of powers property is a very handy shortcut for working with exponents. When you multiply exponentials with the same base, you simply add their exponents. This is because of the properties of multiplication. Consider the expression

• Step 1: Identify the like bases in your expression. For example, in the expression \(7^{-6} \times 7^{-3}\), the common base is 7.
• Step 2: Add the exponents only if the base is the same, following the rule \(a^m \times a^n = a^{m+n}\). For our case, this means adding \(-6\) and \(-3\) to get \(-9\).

This method simplifies expressions significantly, avoiding tedious multiplications or lengthy calculations.

Negative Exponents

When you see a negative exponent, it actually means you are dealing with the reciprocal of the base raised to the corresponding positive exponent. This allows for easier manipulation of terms with exponents.

• Negative powers indicate that the base number is on the wrong side of a fraction. For example, \(a^{-n} = \frac{1}{a^n}\). This makes expressions neater and more convenient to work with.
• To convert a negative exponent to a positive one, simply write the reciprocal of the base with a positive exponent. In the exercise, \(7^{-9}\) can be rewritten as \(\frac{1}{7^9}\).

Understanding negative exponents is crucial in simplifying expressions to their most understandable form.

Algebraic Expressions

Algebraic expressions with exponents can look complex, but they follow predictable rules. Understanding these rules helps in breaking down and simplifying expressions.

• An algebraic expression consists of numbers, variables, and exponents. Operations such as addition, subtraction, multiplication, and division apply.
• In the context of exponents, expressions can be simplified by applying rules like the product of powers and managing negative exponents.
• The exercise clearly shows how you can consolidate expressions with the same base using the product of powers, which results in a cleaner algebraic representation.

Working with algebraic expressions often involves identifying patterns, using properties of exponents, and transforming expressions into forms that are easier to use for further applications.