Problem 34 - Elementary Algebra

Question

Use the product rule and quotient rule of exponents to simplify the following problems. Assume that all bases are nonzero and that all exponents are whole numbers. $$ a^{9} a^{7} $$

Step-by-Step Solution

Answer

Question: Simplify the expression \(a^{9} a^{7}\) using the product rule of exponents. Answer: \(a^{16}\)

Step 1: Identify the base and exponents

In the expression \(a^{9} a^{7}\), we have the same base "a" and two exponents, 9 and 7.

Step 2: Apply the product rule

According to the product rule, when we multiply two expressions with the same base, we add the exponents. Therefore, we have: $$ a^{9} a^{7} = a^{(9+7)} $$

Step 3: Simplify the exponent

Now, let's add the exponents 9 and 7: $$ a^{(9+7)} = a^{16} $$ So, the simplified expression is \(a^{16}\).

Key Concepts

Product Rule of Exponents

When you encounter multiplication involving exponents, the product rule of exponents comes into play. This rule states that if you multiply two powers with the same base, you simply add the exponents. Consider the base to be your anchor; it stays unchanged while the exponents combine to create a new power.

For example, when simplifying an expression like \( a^{9} \times a^{7} \), you'll use the product rule to combine the exponents. This results in \( a^{(9+7)} = a^{16} \), showing that multiplication of exponents is a matter of combining their powers. This technique is essential for simplifying expressions with multiple exponential terms.

Quotient Rule of Exponents

The quotient rule of exponents is used when dividing expressions with exponents and the same base. In this case, instead of adding exponents like the product rule, you subtract the exponent of the denominator from the exponent of the numerator.

Suppose you have \( \frac{a^{9}}{a^{4}} \). Applying the quotient rule, you subtract 4 from 9, giving you \( a^{(9-4)} = a^{5} \). Simplification using the quotient rule helps reduce expressions to their simplest form, making them easier to work with and understand.

Simplifying Expressions

Simplifying expressions is a fundamental skill in algebra. It involves reducing an equation or expression to its simplest form, which makes it easier to interpret or further manipulate. This process may include applying the product and quotient rules of exponents, as well as combining like terms and factoring.

As shown in the exercise with \( a^{9} a^{7} \), the goal was to simplify the expression. By applying the product rule, the expression is simplified to \( a^{16} \), demonstrating a clean and clear calculation without changing the expression's value. Simplification is a critical step that ensures problems are approachable and solvable.

Exponent Properties

Understanding exponent properties is crucial when working with expressions involving powers. These properties not only include the product and quotient rules but also the power of a power rule and the power of a product rule. A comprehensive grasp of these properties allows for efficient simplification of complex algebraic expressions.

For instance, the power of a power rule states that when you raise an exponent to another exponent, you multiply the exponents. If you encounter \( (a^{3})^{2} \), you would calculate this as \( a^{(3 \times 2)} = a^{6} \). Remembering these exponent properties can greatly assist in simplifying and solving more challenging mathematical equations.