Problem 115 - 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 $$ x^{14}+x^{23} $$

Step-by-Step Solution

Answer

The operation is 'None of these'; option e applies here.

Step 1: Identify Each Expression

We have the expression \(x^{14} + x^{23}\). In this context, we are dealing with two separate exponential terms that are being summed together.

Step 2: Determine Operation for Addition

In mathematics, when you add two terms like \(x^{14} + x^{23}\), it means you are simply combining these terms without changing the exponents. There's no standard operation of adding exponents in this case.

Step 3: Identify the Correct Choice

The expression involves addition of terms with different exponents but they cannot be combined by any exponent rule given. Hence, this should be matched with option e: None of these.

Key Concepts

Addition of Exponents

When dealing with exponents, it's essential to understand how they interact during mathematical operations. In scenarios involving addition, such as in the expression \(x^{14} + x^{23}\), each term is treated as an individual component. Unlike coefficients, exponents in addition do not simply add together.

Here's why this is the case: when you see \(x^{14}\) and \(x^{23}\), each part is a unique entity representing a large number multiplied by itself many times. Therefore, even though we are adding these terms together, the exponents (14 and 23) remain unchanged.

To summarize, during the addition of exponents:

• Add like base terms, but keep exponents separate.
• If exponents differ, the terms remain distinct and cannot be combined through addition.

Simplifying Expressions

Simplifying expressions involves breaking down complex terms into their simplest forms while maintaining their original values. In cases like \(x^{14} + x^{23}\), simplification pertains to understanding whether terms can be combined or written in a more compact form.

Here are some key points to remember:

• Identify if the terms share the same base and exponents. If they do, they can often be combined. However, if the exponents differ, they stand alone.
• Assess whether any factoring or common factors exist that can simplify the expression further.
• In our example, since the exponents are different, the expression is already in its simplest form.

By grasping these basics, you'll be better equipped to simplify a wide range of mathematical expressions.

Mathematical Operations

Mathematical operations encompass various actions like addition, subtraction, multiplication, and division. When exponents are involved, operations have specific rules:

1. **Addition and Subtraction**: These operations do not affect the exponents. Each term retains its exponent unless the exponents and bases match, which isn't the case for \(x^{14} + x^{23}\).

2. **Multiplication and Division**: - If multiplying similar bases, you add the exponents (e.g., \(x^a \times x^b = x^{a+b}\)). - If dividing, subtract the exponents (e.g., \(\frac{x^a}{x^b} = x^{a-b}\)).

Understanding these operation rules is vital for tackling expressions in algebra. Whether you are simplifying terms or updating expressions, these foundational rules ensure accurate computation.