Problem 35 - Algebra A Combined Function

Question

Use the power rule and the power of a product or quotient rule to simplify eachexpression. $$ \left(x^{9}\right)^{4} $$

Step-by-Step Solution

Answer

The simplified expression is \(x^{36}\).

Step 1: Identify the Power Rule

The power rule states that for an expression of the form \((a^m)^n\), you can simplify it as \(a^{m \cdot n}\). In this exercise, the expression \((x^{9})^{4}\) is a candidate for the power rule because it is a power of a power.

Step 2: Apply the Power Rule

Apply the power rule to the expression \((x^{9})^{4}\). According to the rule, multiply the exponents: \(9 \times 4 = 36\).

Step 3: Simplify the Expression

Raise \(x\) to the power of 36, which results in the expression \(x^{36}\). This is the simplified form of the original problem.

Key Concepts

Understanding Exponentiation

Exponentiation is a mathematical operation that involves raising a base number to the power of an exponent. This operation is a shorthand way of expressing repeated multiplication of a number by itself. For example, when you see an expression like \(x^9\), it means \(x\) multiplied by itself 9 times: \(x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x \cdot x\).

Exponentiation is essential in algebra because it simplifies how we express large products. Instead of writing out all the multiplications, exponentiation gives us a compact and easier-to-manage notation. When dealing with powers of a power, like in the expression \((x^9)^4\), understanding the rules of exponentiation, such as the power rule, becomes crucial for simplification.

Exploring Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operation symbols that together represent mathematical objects. Expressions like \((x^9)^4\) are built from variables and constants, linked by operations like exponentiation and multiplication.

In our expression, \(x\) is the base, and the numbers are exponents. Algebraic expressions can range from simple to complex, based on the number and type of operations involved. Understanding how to manipulate these expressions using rules like the power rule is vital in algebra. This helps in breaking down complex expressions into manageable pieces to simplify and solve equations effectively.

• Variables: Symbols like \(x\), which can represent any number.
• Constants: Specific numbers that have fixed values, like \(9\) in \(x^9\).
• Operations: Processes like addition, subtraction, multiplication, division, and exponentiation used to form expressions.

Simplifying Expressions Using the Power Rule

Simplifying expressions is an important skill in algebra that involves making expressions as concise as possible without changing their values. The power rule is a key tool for simplification when dealing with powers raised to another power.

For the expression \((x^9)^4\), the power rule states that if you have a power of a power, you multiply the exponents. Here, you would multiply 9 by 4, which equals 36, resulting in \(x^{36}\). This simplification makes the expression easier to work with while retaining its original value.

• Simplified forms are easier to read and to use in equations or further computations.
• Simplifying helps in identifying the core elements of an expression which can be useful for solving algebraic problems.

By applying these simplification techniques, you become skilled in transforming complex expressions into more straightforward forms, which is essential for mastering algebra.