Problem 77 - Algebra A Combined Function

Question

Simplify each expression. $$ a^{2} a^{3} a^{4} $$

Step-by-Step Solution

Answer

The expression simplifies to \( a^9 \).

Step 1: Identify the Expression for Simplification

We need to simplify the expression \( a^2 \cdot a^3 \cdot a^4 \). All the terms are powers of the same variable \( a \).

Step 2: Apply the Laws of Exponents

Use the law of exponents which states that when multiplying like bases, you add their exponents. Thus: \[ a^2 \cdot a^3 \cdot a^4 = a^{2+3+4} \]

Step 3: Perform the Addition in the Exponents

Add the exponents together: \[ 2 + 3 + 4 = 9 \]

Step 4: Write the Simplified Expression

After simplifying the exponents, the expression becomes: \[ a^9 \]

Key Concepts

Exponentiation

Exponentiation is a fundamental operation in mathematics where a number, known as the base, is multiplied by itself a certain number of times, indicated by the exponent. In the expression \(a^2\), \(a\) is the base and 2 is the exponent, meaning \(a\times a\). This operation is a shorthand to simplify repeated multiplication. For example, \(a^3\) is equivalent to \(a \times a \times a\). Exponentiation is not only limited to whole numbers; fractions and decimals can also serve as exponents, broadening the scope of this operation immensely. Understanding exponentiation helps in grasping more complex mathematical concepts, as it creates a foundation for handling large numbers and understanding growth patterns, such as exponential growth. It's crucial in subjects ranging from algebra and calculus to areas like computing and physics.

Simplifying Expressions

Simplifying expressions in mathematics means reducing them to their most concise form without altering their value. For exponential expressions, simplifying often involves using the laws of exponents to combine terms. For instance, the expression \(a^2 \cdot a^3 \cdot a^4\) can be simplified by utilizing the property that when multiplying exponential terms with the same base, you add the exponents:

• Identify common bases: Each term \(a^2\), \(a^3\), and \(a^4\) has the base \(a\).
• Apply the law of exponents: Add the exponents together: \(2 + 3 + 4\).
• Rewrite the expression: The simplified form is \(a^9\).

Simplifying expressions is essential, as it helps to reduce complexity and makes calculations easier, especially when dealing with algebraic equations.

Mathematical Notation

Mathematical notation is the language of mathematics, designed to express mathematical ideas and relationships in a precise and concise way. It includes symbols, expressions, and equations that convey complex information succinctly. For example, the notation \(a^2\) represents the operation of exponentiation, which effectively replaces several multiplication steps with a single compact symbol.Using proper notation is important because:

• It helps in communicating mathematical concepts clearly and precisely.
• It reduces the difficulty in understanding complex equations.
• It aids in avoiding ambiguity, ensuring that everyone interprets the expressions in the same manner.

In simplifying exponential expressions, mathematical notation allows us to denote intricate steps, such as \(a^2 \cdot a^3 \cdot a^4 = a^{2+3+4} = a^9\), which simplifies the communication and understanding of the operations involved.