Problem 32 - College Algebra

Question

Simplify each exponential expression in Exercises 23–64. $$\left(x^{11}\right)^{5}$$

Step-by-Step Solution

Answer

The simplified form of \(\left(x^{11}\right)^{5}\) is \(x^{55}\).

Step 1: Identifying the Exponents

Here, we have \((x^{11})^{5}\), which is in the form \((a^m)^n\). So in this case, \(a = x\), \(m = 11\), and \(n = 5\).

Step 2: Applying the Rule of Exponents

Applying the rule of exponents where \(a^{mn}\), we multiply the exponents together. So, we calculate 11 * 5.

Step 3: Simplifying the Expression

After multiplying the exponents we get \(x^{55}\), which is the simplified form of the exponential expression.

Key Concepts

Rules of Exponents

The rules of exponents are like a toolbox that helps us to handle exponential expressions with ease. These rules tell us how to multiply, divide, and simplify expressions that have powers. For our exercise, we used a specific rule known as the **Power of a Power Rule**. This rule states that when you raise a power to another power, you multiply the exponents. For example, in the expression \((x^{11})^{5}\), the base is \(x\) and we have two exponents: 11 and 5.

• Power of a Power Rule: \((a^m)^n = a^{m \cdot n}\)

Using this rule allows us to combine exponents in a straightforward manner. You simply multiply the exponents (11 and 5 in our case) to get the new exponent, which is \(x^{55}\). Understanding this rule makes working with exponents much less daunting and allows you to simplify problems quickly.

Simplifying Exponents

Simplifying exponents is all about making an expression as concise and manageable as possible. It involves applying the rules of exponents to reduce expressions into simpler forms. When dealing with expressions like \((x^{11})^{5}\), simplifying means transforming it into a form that is easier to understand and work with, in this case, \(x^{55}\).

• Identify the base and the powers involved in the expression.
• Use the appropriate exponent rules to reduce the expression.
• Multiply exponents when using the power of a power rule.

This process not only makes expressions neater but also simplifies computations. As you practice these steps, you’ll become more adept at breaking down even the most complex-looking exponential expressions into their simplest forms.

Algebraic Expressions

In algebra, an expression is a combination of numbers, letters, and operation signs. An algebraic expression becomes more interesting when it involves exponents, like \(x^{11}\). Here, exponents tell us how many times to multiply the base by itself. When they're enclosed in parentheses as in \((x^{11})^{5}\), it means you need to apply the exponents' rules like the power of a power to simplify.

• An algebraic expression can have constants, coefficients, variables, and exponents.
• Using algebra, you can apply operations like addition, subtraction, multiplication, and division to these expressions.
• Exponential algebraic expressions follow specific rules that help simplify and solve them.

Breaking down these expressions using exponent rules is part of the broader skill set you develop in algebra, ensuring you can approach both simplifying and solving a range of mathematical problems effectively.