Question

\(x^{3} x^{4}=?\) \(\begin{array}{lll}\text { (A) } & x^{7} & \text { OR }\end{array}\) (B) \(x^{12}\)

Short answer

The answer is \(x^7\).

Step-by-step solution

Understand the Problem

We are given the expression \(x^3 x^4\) and are asked to simplify it using exponent rules. We must choose between two options: \(x^7\) or \(x^{12}\).

Apply the Product of Powers Rule

The product of powers rule states that when multiplying two exponents with the same base, you add the exponents: \(x^m \times x^n = x^{m+n}\). Here, we apply this rule to \(x^3\) and \(x^4\).

Add the Exponents

Add the exponents from \(x^3\) and \(x^4\): \(3 + 4 = 7\). Therefore, \(x^3 x^4 = x^{3+4} = x^7\).

Choose the Correct Answer

The simplified form of \(x^3 x^4\) is \(x^7\). This matches the option (A). Hence, the correct answer is \(x^7\).

Key concepts

Product of Powers Rule

The Product of Powers Rule is a fundamental principle in mathematics that helps simplify expressions involving exponents. It's particularly useful when you're multiplying terms that have the same base. The rule states that when you multiply two exponents with the same base, you add the exponents together. This can be represented mathematically as follows:

• If you have an expression like \(a^m \times a^n\), you can simplify this to \(a^{m+n}\).

In our exercise, we have two expressions, \(x^3\) and \(x^4\), which share the same base \(x\). According to the Product of Powers Rule, you simply add the exponents:

• The exponents are 3 and 4.
• So, \(3 + 4 = 7\).
• This means \(x^3 \times x^4 = x^7\).

It is crucial to remember that the base must remain the same to use this rule effectively. By understanding this rule, you can easily tackle similar algebraic expressions.

Simplifying Expressions

Simplifying expressions is a technique that helps you present mathematical expressions in their simplest form. This process involves reducing expressions to make them easier to understand and solve. In the given exercise, simplifying is key.

• We started with an expression \(x^3 \times x^4\).
• By using the Product of Powers Rule, we simplified \(x^3 \times x^4\) to \(x^7\).

Why simplify?

• It makes it easier to work with expressions in later steps of a problem.
• It helps in recognizing patterns and relationships between different mathematical concepts.
• It saves time and reduces the risk of making errors in calculation.

By understanding the process of simplification, you can approach algebraic problems with greater confidence and clarity.

Algebraic Expressions

Algebraic expressions play a crucial role in mathematics, especially in algebra. They consist of symbols, numbers, and operators (like addition, subtraction, multiplication, division) structured in a way that represents a specific value or set of values. In our exercise, "\(x^3 \times x^4\)" is an example of an algebraic expression.The core components of algebraic expressions include:

• **Variables:** Symbols like \(x\) or \(y\) that represent unknown values.
• **Constants:** Numbers like 3 or 4 in the expression \(x^3 \times x^4\), which do not change.
• **Operators:** Signs like multiplication (\(\times\)) that dictate how the terms are connected.
• **Exponents:** The small number that writes next to a variable indicating how many times the variable is multiplied by itself, such as the "3" in \(x^3\) or "4" in \(x^4\).

Understanding algebraic expressions is essential for solving algebra problems. They allow you to create models of real-world situations and provide the tools needed to manipulate equations and inequalities effectively. With practice, algebraic expressions become a powerful tool in solving more complex mathematical problems.