Problem 3 - Algebra 2 and Trigonometry

Question

Simplify each expression. In each exercise, all variables are positive. \(x^{3} \cdot x^{4}\)

Step-by-Step Solution

Answer

The simplified expression is \(x^7\).

Step 1: Understand the Properties of Exponents

The problem asks us to simplify the expression \(x^3 \cdot x^4\). The first step is recognizing that when you multiply powers with the same base, you can add their exponents. This follows the property \(a^m \cdot a^n = a^{m+n}\).

Step 2: Apply the Multiplication of Exponents Rule

Using the rule from Step 1, we add the exponents of the same base \(x\), giving us: \(x^3 \cdot x^4 = x^{3+4}\).

Step 3: Simplify the Expression

Perform the addition of the exponents: \(3 + 4 = 7\). This simplifies our expression to \(x^7\).

Key Concepts

Multiplying Powers with Same Base

When dealing with powers that share the same base, a useful property greatly simplifies the process of multiplication. If you see an expression like \(x^3 \cdot x^4\), both terms share the base "\(x\)". The rule for multiplying powers with the same base states that you should add their exponents.
This property can be expressed as:

• \(a^m \cdot a^n = a^{m+n}\)

With this knowledge, combining two powers like \(x^3\) and \(x^4\) becomes straightforward. You just add the exponents, resulting in \(x^{3+4}\), simplifying further to \(x^7\). This purple rule stays consistent across numbers, variables, or any other powers as long as they share the same base.

Simplifying Expressions

Simplifying expressions is a core skill in algebra that involves reducing expressions to their most basic form without changing their value. When you're given an expression involving multiplication of powers, like \(x^3 \cdot x^4\), simplification involves making it as clean and manageable as possible.

• First, identify any properties of exponents that can be used, like adding exponents when multiplying powers with the same base.
• Next, carry out any arithmetic operations, like addition of exponents, to simplify the expression.
• Finally, ensure that the expression is in its simplest form, with no further operations needed.

By transforming \(x^3 \cdot x^4\) into \(x^7\), it becomes easier to work with this expression in further computations or evaluations.

Addition of Exponents

The addition of exponents is a direct result of the multiplying properties of powers. When you multiply powers with the same base, you add the exponents. Let's dissect this a bit more: An exponent tells you how many times to multiply a number by itself.
For instance:

• \(x^3\) means \(x\) multiplied by itself three times: \(x \cdot x \cdot x\).
• \(x^4\) means \(x\) multiplied by itself four times: \(x \cdot x \cdot x \cdot x\).

So, when you multiply \(x^3\) and \(x^4\), you're effectively multiplying \(x\) by itself a total of seven times, illustrating why the rule \(a^m \cdot a^n = a^{m+n}\) holds.
Understanding the rationale behind adding exponents enables you to confidently simplify expressions involving powers, such as finding \(x^7\) from \(x^3 \cdot x^4\).