Problem 32 - 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. $$ x^{5} x^{4} $$

Step-by-Step Solution

Answer

Answer: \(x^{9}\)

Step 1: Identify the base and exponents

In this expression, the base is x, and the exponents are 5 and 4.

Step 2: Apply the product rule of exponents

According to the product rule, when we multiply two expressions with the same base and different exponents, we add the exponents. So, we have: $$ x^{5} \cdot x^{4} = x^{(5+4)} $$

Step 3: Simplify the exponent

Now, we just need to simplify the exponent by adding the numbers 5 and 4: $$ x^{(5+4)} = x^{9} $$ The simplified expression is \(x^{9}\).

Key Concepts

Exponentiation

Exponentiation is a mathematical operation that involves raising a base, such as \(x\), to a power, which is indicated by an exponent. This operation is key for simplifying expressions and solving algebraic problems. In the expression \(x^5 \cdot x^4\):

• The base is \(x\), meaning it's the number being repeatedly multiplied.
• The exponent tells us how many times to multiply the base by itself.

The operation proceeds from left to right in this case, demonstrating how these exponent rules operate on a common base.

Simplifying Expressions

Simplifying mathematical expressions involves reducing them to their simplest form, making them easier to understand and solve. In expressions with exponents, the product rule is a helpful tool for simplification.
The product rule states:

• For bases that are the same, add their exponents when multiplying.

Applied to the original example, it looks like this:

• Start with \(x^5 \cdot x^4\).
• Combine the exponents: \(5 + 4\).
• The result is \(x^9\), the simplest form of the expression.

By using the product rule, the calculation becomes straightforward and quick, showcasing the elegance of mathematical simplification.

Mathematical Bases

A mathematical base is the root value in expressions with exponents. In the context of exponentiation, understanding the base is essential to correctly applying exponent rules. In the expression \(x^5 \cdot x^4\):

• \(x\) is the base, and it remains constant across the terms.
• Maintaining a consistent base simplifies applying exponent rules like the product rule.

Analyzing an expression involves identifying the base and its importance in maintaining balance and correctness through simplifications. By comprehending the base conceptually, students gain a deeper understanding of how expressions can be manipulated while preserving their integrity.