Problem 114 - Algebra A Combined Function

Question

Solve. See the Concept Checks in this section. Match the expression with the operation needed to simplify each. A letter may be used more than once and a letter may not be used at all. a. Add the exponents. b. Subtract the exponents. c. Multiply the exponents. d. Divide the exponents. e. None of these $$ x^{14} \cdot x^{23} $$

Step-by-Step Solution

Answer

The operation needed is 'Add the exponents' (option a).

Step 1: Identify the Expression

The given expression is \( x^{14} \cdot x^{23} \). This expression involves multiplying two exponential terms with the same base.

Step 2: Use the Law of Exponents

When multiplying exponential expressions with the same base, you add the exponents. This is known as the Product of Powers Property: \( a^m \cdot a^n = a^{m+n} \).

Step 3: Apply the Operation

Since the bases are the same, apply the addition of exponents: \( x^{14} \cdot x^{23} = x^{14 + 23} \).

Step 4: Calculate the Sum of the Exponents

Add the exponents: \( 14 + 23 = 37 \). Therefore, the expression simplifies to \( x^{37} \).

Step 5: Match the Correct Operation

The operation used to simplify is 'Add the exponents'. Therefore, the correct match is option (a).

Key Concepts

Product of Powers Property

The "Product of Powers Property" is a fundamental rule in exponentiation that makes simplifying expressions much easier. This property states that when you multiply two powers that have the same base, you simply add the exponents. It can be mathematically represented as: \[ a^m \cdot a^n = a^{m+n} \]This means if you have a base \( a \), and you multiply \( a^{14} \) by \( a^{23} \), you will combine the exponents to get a single exponent, adding them up to 37 in this case.Here are key points to remember about this property:

• It only applies when the bases are identical.
• The operation of addition is performed on the exponents, not on the bases themselves.
• The base remains unchanged after applying the product of powers property.

Knowing this property helps in quickly simplifying expressions and making calculations more manageable.

Multiplying Exponents

"Multiplying exponents" can sometimes be misleading in the context of exponentiation rules. You do not literally multiply the exponents when using the product of powers property, even though the term might suggest so. Instead, you multiply numbers that are expressed as exponential terms by adding their exponents.For example, in the expression \( x^{14} \cdot x^{23} \), you combine the exponents to become:

• 14 from the first term
• 23 from the second term

Adding these together: \[ 14 + 23 = 37 \]Thus, "multiplying exponents'" correctly implies finding the result of exponential terms multiplication through addition of similar base exponents. This concept plays a crucial role in algebraic manipulations.Notably, when bases are the same, the multiplication of terms is straightforward, avoiding more complex operations like multiplying the base values themselves.

Exponents Simplification

"Exponents simplification" is the process of using rules, like the product of powers property, to make exponentiation expressions easier to manage and interpret.The goal is to reduce complexity in expressions, which aids in performing further operations like addition, subtraction, or even more multiplication if necessary.Consider these simplification techniques:

• Identify if there is a common base shared between exponential terms.
• Use properties such as product of powers, which we've previously discussed.
• Perform operations on exponents (adding, subtracting, etc.) based on rules of exponents.

In the case of \( x^{14} \cdot x^{23} \), the simplification leads directly to a cleaner expression \( x^{37} \). By learning to apply these rules efficiently, you will often find complex algebraic expressions becoming much simpler, paving the way for quicker solutions and better understanding of mathematical problems.