Problem 28 - College Algebra

Question

Simplify each exponential expression in Exercises 23–64. $$x^{11} \cdot x^{5}$$

Step-by-Step Solution

Answer

The simplified form of the given exponential expression is \(x^{16}\)

Step 1: Define the given exponential expression

The given exponential expression is \(x^{11} \cdot x^{5}\)

Step 2: Apply the multiplication of exponential expression rule

According to the multiplication rule for exponents, when multiplying like bases, the exponents should be added. So, \(x^{11} \cdot x^{5} = x^{11+5}\)

Step 3: Solve

Adding the exponents 11 and 5 together, we get \(x^{16}\)

Key Concepts

Exponential Expression Rules

Exponential expressions are a fundamental concept in algebra that involve numbers or variables raised to a power. To simplify these expressions, it is essential to understand the rules that govern how to manipulate exponents. The most basic rules include the product rule, quotient rule, power of a power rule, and zero exponent rule.

With the product rule, when you multiply two exponential expressions that have the same base, you keep the base and add the exponents together, symbolized as \(a^m \cdot a^n = a^{m+n}\). Similarly, the quotient rule states that when you divide exponential expressions with the same base, you subtract the exponent in the denominator from the exponent in the numerator, \(a^m / a^n = a^{m-n}\). When an exponent is raised to another exponent, known as the power of a power rule, you multiply the exponents, \((a^m)^n = a^{m \cdot n}\). Lastly, any base raised to the exponent of zero is equal to one, \(a^0 = 1\), except for the indeterminate form \(0^0\).

Using these rules when faced with an exponential expression simplifies the process of manipulation and ensures accuracy in finding the solution.

Multiplication of Exponents

Understanding the multiplication of exponents is vital for any student studying algebra. This concept, as demonstrated in our exercise with \(x^{11} \cdot x^{5}\), revolves around the product rule. As you may recall, when you multiply two exponents with the same base, you simply add the exponents.

An easy way to remember this rule is to think of the multiplication of exponents as a way of counting how many times you're using the base in repeated multiplication. For instance, \(x^{11}\) means you're multiplying x by itself 11 times, and \(x^{5}\) means you're multiplying x by itself 5 times. So, when you combine \(x^{11} \cdot x^{5}\), you're actually multiplying x by itself 11+5 times, which is where the rule of adding exponents comes from.

It's essential to note that this rule only applies when the bases are the same. If the bases are different, you cannot use this simplification method. Also, remember that this rule applies to any type of base, whether it be numerical or algebraic.

Algebraic Operations

Algebraic operations such as addition, subtraction, multiplication, and division are the building blocks of algebra. In the context of exponential expressions, these operations help us combine and simplify terms to obtain a more manageable equation or expression.

Aside from the multiplication of exponents, addition and subtraction come into play when you're dealing with coefficients (numbers in front of the variables) or when combining like terms. It is crucial to remember that like terms are terms that have the same variable parts raised to the same power. Multiplication and division, as we've seen, involve handling exponents in a specific manner depending on whether you are combining like bases as a product or as a quotient.

Furthermore, these operations have a certain order, known as the order of operations or PEMDAS/BODMAS, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Following this order ensures proper simplification and manipulation of algebraic expressions. With practice and familiarization, performing algebraic operations with confidence becomes second nature.