Problem 27 - Algebra and Trigonometry

Question

Simplify each exponential expression. $$ x^{3} \cdot x^{7} $$

Step-by-Step Solution

Answer

The simplified exponential expression is \( x^{10} \).

Step 1: Identify the same base and its exponents

We are given \( x^{3} \cdot x^{7} \). Here, \( x \) is the base which is same in both terms, and 3 and 7 are the exponents.

Step 2: Apply the rule for multiplying powers with the same base

When multiplying with the same base, we add the exponents: \( x^{3+7} \).

Step 3: Simplify the expression

Adding the exponents 3 and 7, we get \( x^{10} \).

Key Concepts

Exponent Rules

Exponents are the way we express repeated multiplication of a number by itself. They consist of a base and an exponent. In mathematics, we have certain rules that help us manage exponents easily. One of the basic rules we often use is the Product of Powers Property. This rule states that if you multiply matching bases, you simply add the exponents.
For example, if you are multiplying two expressions like \( x^{a} \cdot x^{b} \), you can combine them into \( x^{a+b} \). This rule is crucial because it simplifies our calculations and helps maintain consistency throughout mathematical operations. Understanding and using these exponent rules effectively can make working with algebra a lot more straightforward.
Another critical rule to remember is that any number raised to the power of zero is one. Therefore, \( x^0 = 1 \) for any non-zero \( x \). Mastering these properties allows us to simplify various expressions in algebra and ensures mathematical expressions remain clear and simple to follow.

Simplifying Expressions

Simplifying expressions is an essential skill in algebra. It involves reducing an expression to its simplest form, which makes it easier to understand and work with. When simplifying expressions, we often use exponent rules. By applying rules such as the Product of Powers Property, we can condense expressions into more manageable forms.
Think of simplifying like tidying up a messy room: you make everything neat and easy to access. To simplify an expression like \( x^{3} \cdot x^{7} \), you combine the exponents by adding them, according to the product rule, to get \( x^{10} \). This process of simplification can then make further calculations or other algebraic manipulations more straightforward.
Always look for opportunities to combine like terms, and remember that simplification is about making expressions as clear and concise as possible. It's a stepping stone to solving equations and mastering other concepts in algebra.

Multiplying Powers

When it comes to multiplying powers, there's a straightforward process that involves following the exponent rules. Specifically, when working with the same base in multiplication, you add the exponents together. This neat little process is part of the Product of Powers Property.
For instance, consider the expression \( x^{3} \cdot x^{7} \). Since both terms share the same base, \( x \), you can merge these terms by adding the exponents: \( x^{3+7} \). This results in \( x^{10} \).
The secret to multiplying powers efficiently lies in recognizing shared bases and understanding that the operation boils down to simple addition of the exponents. It's a key skill in algebra that reduces unexpected errors and allows for quick simplification, helping to solve larger algebraic problems with confidence.

Algebra

Algebra is the branch of mathematics dealing with symbols and the rules for manipulating these symbols. It is all about discovering the unknowns. Algebra provides a powerful tool for solving a wide range of problems by using variables, constants, and the operations of addition, subtraction, multiplication, and division.
In simple terms, algebra involves working with equations where letters represent numbers. These can be solved to find the values of these numbers. In the process, you'll likely need to use exponent rules and simplify expressions.
Understanding concepts like multiplying powers is crucial in algebra because expressions often require simplification through combining like terms. The product rule enables you to handle repeated multiplication efficiently, which is common in algebraic equations.
To help with solving algebra problems:

• Identify the operations you need to perform.
• Simplify expressions wherever possible.
• Apply the appropriate rules, such as exponent rules, to get to the solution smoothly.

Remember, mastering algebra starts with grasping the basics and practicing regularly.