Problem 84 - Algebra 2

Question

Simplify. Assume that no variable equals zero. \(x^{4} \cdot x^{6}\)

Step-by-Step Solution

Answer

The expression simplifies to \(x^{10}\).

Step 1: Identify the Rule of Exponents

When you multiply expressions that have the same base, you can simplify by adding the exponents according to the rule: \[ a^m imes a^n = a^{m+n} \]

Step 2: Apply the Rule of Exponents

The base in the expression \(x^4 \cdot x^6\) is \(x\). Apply the rule of exponents by adding the exponents 4 and 6: \[ x^{4+6} \]

Step 3: Simplify the Expression

Add the exponents 4 and 6 to simplify the expression: \[ x^{4+6} = x^{10} \] Thus, the simplified form of \(x^4 \cdot x^6\) is \(x^{10}\).

Key Concepts

Multiplying Powers with the Same Base

Multiplying powers with the same base is a fundamental concept in algebra. When you see expressions like \(x^4 \cdot x^6\), you are looking at two powers with the same base, in this case, \(x\). To simplify such expressions, you need to know a basic rule of exponents:

• If you multiply powers with the same base, you can simplify by adding their exponents.

This is written mathematically as \(a^m \cdot a^n = a^{m+n}\). Here, \(a\) is the base, and \(m\) and \(n\) are the exponents.
So, when you have \(x^4 \cdot x^6\), you add the exponents 4 and 6, which gives you \(x^{4+6}\), resulting in \(x^{10}\). This step doesn't change the base, only combines the power by the rule of exponents.

Simplifying Expressions

Simplifying expressions is a process that makes mathematical expressions easier to understand or work with. This often involves combining like terms or using algebraic rules to restate expressions in a simpler form.
In the case of algebraic expressions like \(x^4 \cdot x^6\), simplifying means using rules for exponents to combine the terms into a more straightforward form. Here, adding the exponents to give \(x^{10}\) is a simplification.

• This process reduces the complexity of calculations.
• Simplification helps in solving equations and inequalities.

By using the rule that when multiplying two exponents with the same base, you add the exponents, you turn a complex-looking multiplication into a simpler power. This makes it easier to work with the expression, especially in more advanced algebraic equations.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations. They can represent patterns, models, or real-world situations and are a core part of algebra.

• Variables like \(x\) represent unknown or changeable values.
• Powers, such as \(x^4\), indicate repeated multiplication of a base.

Understanding how to manipulate these expressions is essential in algebra. In the example \(x^4 \cdot x^6\), the expression consists of two powers of the same base. By understanding the rules for multiplying powers, you simplify these expressions into one term \(x^{10}\).
This process is not only about finding the simplest form but also about gaining the ability to solve algebraic equations more efficiently. Simplified expressions are easier to handle, reducing errors in calculations and making complex problems more manageable.