Problem 77 - Algebra A Combined Function

Question

Simplify, if possible. $$ x^{4} \cdot x^{9} $$

Step-by-Step Solution

Answer

The expression simplifies to \(x^{13}\).

Step 1: Identify the Law of Exponents

The expression involves multiplying two expressions with the same base \(x\). There is a law of exponents that states \(a^{m} \cdot a^{n} = a^{m+n}\). We'll use this rule to simplify the expression.

Step 2: Apply the Law of Exponents

Using the law of exponents \(a^{m} \cdot a^{n} = a^{m+n}\), apply it to the problem at hand. Here the base is \(x\), and the exponents are \(4\) and \(9\). Thus, the expression simplifies to \(x^{4 + 9}\).

Step 3: Perform the Addition

Add the exponents from Step 2: \(4 + 9 = 13\).

Step 4: Write the Simplified Expression

The simplified expression, after applying the law of exponents and calculating the addition, becomes \(x^{13}\).

Key Concepts

Simplifying Expressions

Simplifying expressions is all about making complex mathematical expressions easier to work with. When you simplify, you're trying to consolidate and reduce expressions to their most basic form. In the given exercise, you're dealing with a multiplication of the same base with different exponents — the base being the variable "x" and the exponents being 4 and 9.

To simplify, follow these steps:

• Identify the expressions that can be combined.
• Look for common bases. In this case, both terms have the base "x".
• Use mathematical rules, such as exponent rules, to make calculations easier.

By applying the rules correctly, what seems complex will transform into something much easier to manage. This simplification makes further mathematical operations or understanding the expression's behavior much simpler.

Law of Exponents

The law of exponents is a set of mathematical principles that help manage how exponents behave. These laws are crucial when simplifying expressions with exponents.

One of the main rules you'll use is:
\(a^m \cdot a^n = a^{m+n}\).

• This law states that when you multiply like bases, you simply add the exponents together.
• It is important because it simplifies expressions without having to perform more complex calculations.

In our exercise, the base "x" is consistent, and the rule is easily applied. You don't need to separately multiply 'x' by itself multiple times; simply add the exponents (4 and 9). This gives you the simplified form \(x^{13}\).

Knowing these laws will speed up simplification and enhance overall understanding of algebraic operations.

Multiplying Exponents

Multiplying exponents can seem tricky at first, but it's straightforward with the right approach. The key lies in understanding that you're dealing with repeated multiplication of numbers with the same base.

Here's how multiplying exponents works:

• When two exponential terms with the same base are multiplied, the exponents are added together.
• This is based directly on the law of exponents explained above, \(a^m \cdot a^n = a^{m+n}\).

For example, \(x^4 \cdot x^9\) becomes \(x^{4+9}\) due to exponent multiplication, resulting in \(x^{13}\).

Recognizing patterns in exponent multiplication will make these expressions less daunting and more intuitive. Understanding these patterns is foundational, as it serves as a building block for more complex algebraic manipulations.